From d66562644f8c9b0a82645c740d3fb7b22f7ed95d Mon Sep 17 00:00:00 2001
From: cuppajoeman <ccn@cuppajoeman.com>
Date: Tue, 3 Sep 2024 13:38:28 -0400
Subject: [PATCH] fix(topology): clear up index file content

---
 html/topology/index.html | 708 ---------------------------------------
 1 file changed, 708 deletions(-)

diff --git a/html/topology/index.html b/html/topology/index.html
index 04569b5..508ebfe 100644
--- a/html/topology/index.html
+++ b/html/topology/index.html
@@ -32,714 +32,6 @@
                     <li>Quotient Topology</li>
                 </ul>
             </fieldset>
-
-
-<!--            <div class="definition" id="product-topology">-->
-<!--                <div class="title">-->
-<!--                    product topology-->
-<!--                </div>                 -->
-<!--                <div class="content">-->
-<!--                     Let \( X \) and \( Y \) be topological spaces. The product topology on \( X \times Y \) is the topology having as basis the collection \( B \) of all sets of the form \( U \times V \), where \( U \) is an open subset of \( X \) and \( V \) is an open subset of \( Y \)-->
-<!--                </div>          -->
-<!--            </div>-->
-
-
-
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                if B is a basis for the topology of X and C is a basis for the topology-->
-<!--of Y , then the collection-->
-<!--D = {B × C | B ∈ B and C ∈ C}-->
-<!--is a basis for the topology of X × Y .-->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    so D is a basis for X × Y .-->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition" id="projection-function">-->
-<!--                <div class="title">projection function-->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    Let π1 : X × Y → X be defined by the equation π1(x, y) = x; let π2 : X × Y → Y be defined by the equation π2(x, y) = y.  The maps π1 and π2 are called the projections of X × Y onto its first and second factors, respectively.-->
-<!--                </div>          -->
-<!--            </div>-->
-
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                   The collection-->
-<!--S-->
-<!-- = {π−1-->
-<!--1 (U) | U open in X}∪{π−1-->
-<!--2 (V) | V open in Y }-->
-<!--is a subbasis for the product topology on X × Y .    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    Let X be a topological space with topology T . If Y is a subset of X, the-->
-<!--collection TY = {Y ∩ U | U ∈ T }-->
-<!--is a topology on Y , called the subspace topology. With this topology, Y is called a-->
-<!--subspace of X; its open sets consist of all intersections of open sets of X with Y .   -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    If B is a basis for the topology of X then the collection-->
-<!--BY = {B ∩ Y | B ∈ B}-->
-<!--is a basis for the subspace topology on Y-->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">-->
-<!--                                  -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    Let Y be a subspace of X. If U is open in Y and Y is open in X, then-->
-<!--U is open in X.   -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    f A is a subspace of X and B is a subspace of Y , then the product-->
-<!--topology on A × B is the same as the topology A × B inherits as a subspace of X ×Y     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Let X be an ordered set in the order topology; let Y be a subset-->
-<!--of X that is convex in X. Then the order topology on Y is the same as the topology Y-->
-<!--inherits as a subspace of X.-->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                    A subset A of a topological space X is said to be closed if the set X − A is open.   -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    Let X be a topological space. Then the following conditions hold:-->
-<!--(1) ∅ and X are closed.-->
-<!--(2) Arbitrary intersections of closed sets are closed.-->
-<!--(3) Finite unions of closed sets are closed.    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Let Y be a subspace of X. Then a set A is closed in Y if and only if-->
-<!--it equals the intersection of a closed set of X with Y .    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    Let Y be a subspace of X. If A is closed in Y and Y is closed in X,-->
-<!--then A is closed in X.     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Given a subset A of a topological space X, the interior of A is defined as the union of-->
-<!--all open sets contained in A,     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    the closure of A is defined as the intersection of all-->
-<!--closed sets containing A     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Let Y be a subspace of X; let A be a subset of Y ; let A¯ denote the-->
-<!--closure of A in X. Then the closure of A in Y equals A¯ ∩ Y .    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Let A be a subset of the topological space X.-->
-<!--(a) Then x ∈ A¯ if and only if every open set U containing x intersects A.-->
-<!--(b) Supposing the topology of X is given by a basis, then x ∈ A¯ if and only if every-->
-<!--basis element B containing x intersects A.    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    Mathematicians often use some special terminology here. They shorten the state- -->
-<!--ment “U is an open set containing x” to the phrase-->
-<!--“U is a neighborhood of x.”     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    If A is a subset of the topological space X and if x is a point of X, we say that x is a-->
-<!--limit point (or “cluster point,” or “point of accumulation”) of A if every neighborhood-->
-<!--of x intersects A in some point other than x itself. Said differently, x is a limit point-->
-<!--of A if it belongs to the closure of A − {x}. The point x may lie in A or not; for this-->
-<!--definition it does not matt    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    Let A be a subset of the topological space X; let A be the set of all-->
-<!--limit points of A. Then     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    A subset of a topological space is closed if and only if it contains all-->
-<!--its limit points.     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                     In an-->
-<!--arbitrary topological space, one says that a sequence x1, x2, ... of points of the space-->
-<!--X converges to the point x of X provided that, corresponding to each neighborhood U-->
-<!--of x, there is a positive integer N such that xn ∈ U for all n ≥ N.     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                    A topological space X is called a Hausdorff space if for each pair x1, x2-->
-<!--of distinct points of X, there exist neighborhoods U1, and U2 of x1 and x2, respectively,-->
-<!--that are disjoint.    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">   -->
-<!--                    Every finite point set in a Hausdorff space X is closed   -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Let X be a space satisfying the T1 axiom; let A be a subset of X.-->
-<!--Then the point x is a limit point of A if and only if every neighborhood of x contains-->
-<!--infinitely many points of A.     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">   -->
-<!--                    If X is a Hausdorff space, then a sequence of points of X converges-->
-<!--to at most one point of X.   -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                    Every simply ordered set is a Hausdorff space in the order topology.-->
-<!--The product of two Hausdorff spaces is a Hausdorff space. A subspace of a Hausdorff-->
-<!--space is a Hausdorff space.    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">continuous function</div>-->
-<!--                <div class="content"> -->
-<!--                    Let \( X \) and \( Y \) be topological spaces. A function \( f{:} X \to Y \) is said to be continuous if-->
-<!--for each open subset \( V \) of \( Y \), the set \( f{−} 1 \left ( V \right ) \) is an open subset of \( X \).     -->
-<!--                </div>          -->
-<!--            </div>-->
-
-<!--            <div class="exercise">-->
-<!--                <div class="title"></div>-->
-<!--                <div class="content">-->
-<!--                    prove that this definition of continuity is equivalent to the <span class="knowledge-link" data-href="/analysis/single_variable.html#definition-limit-of-a-function-at-a-point">epsilon delta definition</span>-->
-<!--                </div>-->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition" id="continuous-function-for-a-basis">-->
-<!--                <div class="title">continous function for a basis</div>-->
-<!--                <div class="content"> -->
-<!--                    given \( f{:} X \to Y \) if the topology of the range space Y is given by a basis \( \mathcal{B} \), then to prove continuity of \( f \) it suffices to show that the inverse image of every basis element is open-->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">-->
-<!--                    Let X and Y be topological spaces; let f : X → Y . Then the-->
-<!--following are equivalent:-->
-<!--(1) f is continuous.-->
-<!--(2) For every subset A of X, one has f (A¯) ⊂ f (A).-->
-<!--(3) For every closed set B of Y , the set f −1(B) is closed in X.-->
-<!--(4) For each x ∈ X and each neighborhood V of f (x), there is a neighborhood U-->
-<!--of x such that f (U) ⊂ V.-->
-<!--If the condition in (4) holds for the point x of X, we say that f is continuous at-->
-<!--the point x.      -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition" id="homeomorphism">-->
-<!--                <div class="title">homeomorphism</div>-->
-<!--                <div class="content">-->
-<!--                    Let \( X \) and \( Y \) be topological spaces; let \( f{:} X \to Y \) be a bijection. If both the function \( f \) and \( f^{- 1}{:} Y \to X \) are continuous, then \( f \) is called a homeomorphism-->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition" id="imbedding">-->
-<!--                <div class="title">imbedding</div>-->
-<!--                <div class="content"> -->
-<!--                    Now suppose that \( f{:} X → Y \) is an injective continuous map, where \( X \) and \( Y \) are topological spaces. Let \( Z \) be the image set \( f{\left ( X \right )} \), considered as a subspace of \( Y \) ; then the function \( f |_{Z} : X → Z \) obtained by restricting the range of \( f \) is bijective. If \( f |_{Z} \) happens to be a homeomorphism of \( X \) with \( Z \), we say that the map \( f{:} X → Y \) is a topological imbedding, or simply an imbedding, of \( X \) in \( Y \) .     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            -->
-<!--            <div class="theorem" id="rules-for-constructing-continuous-functions">-->
-<!--                <div class="title">             -->
-<!--                    rules for constructing continuous functions-->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                     Let \( X , Y \) and \( Z \) be topological spaces, the the following statements all hold:-->
-<!--                     <ol>-->
-<!--                        <li>-->
-<!--                            (Constant function) If \( f{:} X → Y \) maps all of \( X \) into the single point \( y_{0} \) of \( Y \) then \( f \) is continuous-->
-<!--                        </li>-->
-<!--                        <li>-->
-<!--                            (Inclusion) If \( A \) is a subspace of \( X \), the inclusion function \( j : A → X \) is continuous-->
-<!--                        </li>-->
-<!--                        <li>-->
-<!--                            (Composites) If \( f{:} X → Y \) and \( g{:} Y → Z \) are continuous, then the map \( g{\circ} f{:} X → Z \) is continuous.-->
-<!--                        </li>-->
-<!--                        <li>-->
-<!--                            (Restricting the domain) If \( f{:} X → Y \) is continuous, and if \( A \) is a subspace of \( X \), then the restricted function \( f |_{A} : A → Y \) is continuous.-->
-<!--                        </li>-->
-<!--                        <li>-->
-<!--                            (Restricting or expanding the range) Let \( f{:} X → Y \) be continuous. If \( Z \) is a subspace of \( Y \) containing the image set \( f{\left ( X \right )} \), then the function \( g{:} X \to Z \) obtained by restricting the range of \( f \) is continuous. If \( Z \) is a space having \( Y \) as a subspace, then the function \( h : X \to Z \) obtained by expanding the range of \( f \) is continuous-->
-<!--                        </li>-->
-<!--                        <li>-->
-<!--                            (Local formulation of continuity) The map \( f{:} X \to Y \) is continuous if \( X \) can be written as the union of open sets \( \bigcup_{\alpha} \) such that \( f |_{\bigcup_{\alpha}} \) is continuous for each \( \alpha \)-->
-<!--                        </li>-->
-<!--                     </ol>-->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Let X = A ∪ B, where A and B are closed-->
-<!--in X. Let f : A → Y and g : B → Y be continuous. If f (x) = g(x) for every-->
-<!--x ∈ A ∩ B, then f and g combine to give a continuous function h : X → Y , defined-->
-<!--by setting h(x) = f (x) if x ∈ A, and h(x) = g(x) if x ∈ B.     -->
-<!--                </div>          -->
-<!--            </div>-->
-
-<!--            <div class="lemma" id="projection-functions-are-continuous">-->
-<!--                <div class="title">projection functions are continuous</div>-->
-<!--                <div class="content">-->
-<!--                    The <a href="#projection-function">projection functions</a> \( \pi_{1} : X \times Y \to X \) and \( \pi_{2} : X \times Y \to Y \) are continuous -->
-<!--                </div>-->
-<!--            </div>-->
-
-<!--            <div class="proof">-->
-<!--                <div class="title"></div>-->
-<!--                <div class="content">-->
-<!--                    <p>-->
-<!--                        We'll start by showing that \( \pi_{1} \) is continuous, so let \( U \) be open in \( X \) and now we can note that \( \pi_{1}^{- 1} \left ( U \right ) = U \times Y \) since the second component never matters in the computation of \( \pi_{1} \left ( x , y \right ) = x \)-->
-<!--                    </p>-->
-<!--                    <p>-->
-<!--                        By the very definition of the <a href="#product-topology">product toplogy</a>, we can see it is generated by the basis consisting of \( U \times V \) open in \( X \) and \( Y \) respectively, we assumed \( U \) was open, \( Y \) is open in itself by the definition of toplogy, thus \( U \times Y \) is open in \( X \times Y \) since <a href="#basis-elements-are-open">every basis element is also in the toplogy</a>. -->
-<!--                    </p>-->
-<!--                    <p>-->
-<!--                        Thus we may conclude that \( \pi_{1} \) is continuous, the proof for \( \pi_{2} \) may be obtained symmetrically.-->
-<!--                    </p>-->
-<!--                </div>-->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="theorem" id="product-functions-are-continuous">-->
-<!--                <div class="title">                 -->
-<!--                    product functions are continuous-->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                    Let \( f{:} A \to X \times Y \) be given by the equation \( f{\left ( a \right )} = \left ( f_{1}{\left ( a \right )} , f_{2}{\left ( a \right )} \right ) \). Then \( f \) is continuous if and only if the functions \( f_{1}{:} A \to X \) and \( f_{2}{:} A \to Y \) are continuous. The maps \( f_{1} \) and \( f_{2} \) are called the coordinate functions of \( f \)    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-
-
-<!--            <div class="exercise">-->
-<!--                <div class="title">winking functions are continuous</div>-->
-<!--                <div class="content">-->
-<!--                   Given \( x_{0} \in X \) and \( y_{0} \in Y \), show that the maps \( f{:} X \to X \times Y \) and  \( g{:} Y \to X \times Y \) defined by-->
-<!--                   <div class="centered-content">-->
-<!--                        \( f{\left ( x \right )} = x \times y_{0} \) and \( g{\left ( y \right )} = x_{0} \times Y \)-->
-<!--                   </div>-->
-<!--                   are <a href="#imbedding">imbeddings</a>.-->
-<!--                </div>-->
-<!--            </div>-->
-
-<!--            <div class="proof">-->
-<!--                <div class="title"></div>-->
-<!--                <div class="content">-->
-<!--                    <details>-->
-<!--                        <summary>details</summary>-->
-
-<!--                        <p>-->
-<!--                            We'd like to show that \( f \) is an <a href="#imbedding">imbedding</a>, that is that both \( f^{'} \) and \( \left ( f^{'} \right )^{- 1} \) are both continuous functions where \( f{'} \) is \( f \) with it's range restricted to \( f{\left ( X \right )} \).-->
-<!--                        </p>-->
-
-<!--                        <p>-->
-<!--                            Starting with \( f^{'} \) and acutally looking at the original \( f \) we can see that it can be defined as \( f{\left ( x \right )} = \left ( i d \left ( x \right ) , c \left ( x \right ) \right ) \) with the identity and constant function respectively. Looking at the <a href="#rules-for-constructing-continuous-functions">rules for contructing continuous functions</a>, we can see that \( i d : X \to X \) is simply an inclusion function from \( X \) into itself, and that \( c : Y \to y_{0} \) is a constant funciton, and thus are both continuous, also since <a href="#product-functions-are-continuous">product functions are continuous</a> we know that \( f \) is continuous, using rule 4, we can see that \( f{'} \) is also continuous.-->
-<!--                        </p>-->
-
-<!--                        <p>-->
-<!--                            Looking at \( \left ( f^{'} \right )^{- 1} : X \times \left \lbrace y_{0} \right \rbrace \to X \), we can see that this function simply drops the second component and thus \( \left ( f^{'} \right )^{- 1} = \pi_{1} \) and since <a href="projection-functions-are-continuous">projection functions are continuous</a>, we've shown that the inverse function is continuous.-->
-<!--                        </p>-->
-
-<!--                        <p>-->
-<!--                            With all that said, we now know that \( f{'} \) is a homeomorphism, which by definition means that \( f \) is an imbedding, which is what we needed to show.-->
-<!--                        </p>-->
-<!--                    </details>-->
-
-
-
-
-<!--                </div>-->
-<!--            </div>-->
-
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                    Let J be an index set. Given a set X, we define a J -tuple of elements-->
-<!--of X to be a function x : J → X. If α is an element of J , we often denote the value-->
-<!--of x at α by xα rather than x(α); we call it the αth coordinate of x. And we often-->
-<!--denote the function x itself by the symbol-->
-<!--(xα)α∈J ,-->
-<!--which is as close as we can come to a “tuple notation” for an arbitrary index set J . We-->
-<!--denote the set of all J -tuples of elements of X by X J .    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                    Let {Aα}α∈J be an indexed family of sets; let X = -->
-<!--α∈J Aα. The-->
-<!--cartesian product of this indexed family, denoted by-->
-<!---->
-<!--α∈J-->
-<!--Aα,-->
-<!--is defined to be the set of all J -tuples (xα)α∈J of elements of X such that xα ∈ Aα for-->
-<!--each α ∈ J . That is, it is the set of all functions-->
-<!--x : J → - -->
-
-<!--α∈J-->
-<!--Aα-->
-<!--such that x(α) ∈ Aα for each α ∈ J .    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                    Let {Xα}α∈J be an indexed family of topological spaces. Let us take as-->
-<!--a basis for a topology on the product space-->
-<!---->
-<!--α∈J-->
-<!--Xα-->
-<!--the collection of all sets of the form-->
-<!---->
-<!--α∈J-->
-<!--Uα,-->
-<!--where Uα is open in Xα, for each α ∈ J . The topology generated by this basis is called-->
-<!--the box topology.    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                     Let-->
-<!--πβ :-->
-<!---->
-<!--α∈J-->
-<!--Xα → Xβ-->
-<!--be the function assigning to each element of the product space its βth coordinate,-->
-<!--πβ((xα)α∈J ) = xβ;-->
-<!--it is called the projection mapping associated with the index β.     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                    Let - -->
-<!--β denote the collection-->
-<!-- - -->
-<!--β = {π−1-->
-<!--β (Uβ) | Uβ open in Xβ},-->
-<!--and let - -->
-<!-- denote the union of these collections,-->
-<!-- - -->
-<!-- = - -->
-
-<!--β∈J-->
-<!-- - -->
-<!--β.-->
-<!--The topology generated by the subbasis - -->
-<!-- is called the product topology. In this topol- -->
-<!--ogy -->
-
-<!--α∈J Xα is called a product space.    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">   -->
-<!--                    The box topol- -->
-<!--ogy on -->
-<!-- Xα has as basis all sets of the form -->
-<!--Uα, where Uα is open in Xα for-->
-<!--each α. The product topology on-->
-<!-- Xα has as basis all sets of the form-->
-<!--Uα, where-->
-<!--Uα is open in Xα for each α and Uα equals Xα except for finitely many values of α   -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">   -->
-<!--                    Suppose the topology on each space Xα is given by a basis Bα. The-->
-<!--collection of all sets of the form-->
-<!---->
-<!--α∈J-->
-<!--Bα,-->
-<!--where Bα ∈ Bα for each α, will serve as a basis for the box topology on-->
-
-<!--α∈J Xα.-->
-<!--The collection of all sets of the same form, where Bα ∈ Bα for finitely many-->
-<!--indices α and Bα = Xα for all the remaining indices, will serve as a basis for the-->
-<!--product topology-->
-
-<!--α∈J Xα.   -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">   -->
-<!--                    Let Aα be a subspace of Xα, for each α ∈ J . Then -->
-<!-- Aα is a-->
-<!--subspace of-->
-<!-- Xα if both products are given the box topology, or if both products are-->
-<!--given the product topology.   -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">   -->
-<!--                     If each space Xα is Hausdorff space, then-->
-<!-- Xα is a Hausdorff space-->
-<!--in both the box and product topologies.   -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Let {Xα} be an indexed family of spaces; let Aα ⊂ Xα for each α. If-->
-<!--Xα is given either the product or the box topology, then-->
-<!-- A¯α =  Aα.     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Let f : A → -->
-
-<!--α∈J Xα be given by the equation-->
-<!--f (a) = ( fα(a))α∈J ,-->
-<!--where fα : A → Xα for each α. Let -->
-<!-- Xα have the product topology. Then the-->
-<!--function f is continuous if and only if each function fα is continuous     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition" id="definition-metric-on-a-set">-->
-<!--                <div class="title">metric on a set</div>-->
-<!--                <div class="content">  -->
-<!--                     A metric on a set \( X \) is a function \( d : X \times X \to \mathbb{R} \) having the following properties:-->
-<!--                     <ol>-->
-<!--                        <li>\( d \left ( x , y \right ) \ge 0 \) for all \( x , y \in X \); equality holds if and only if \( x = y \)</li>-->
-<!--                        <li>\( d \left ( x , y \right ) = d \left ( y , x \right ) \) for all \( x , y \in X \)</li>-->
-<!--                        <li>\( d \left ( x , y \right ) + d \left ( y , z \right ) \ge d \left ( x , z \right ) \) for all \( x , y , z \in X \)</li>-->
-<!--                     </ol>-->
-<!--                    Given a metric \( d \) on \( X \), the number \( d \left ( x , y \right ) \) is often called the distance between \( x \) and \( y \)    -->
-<!--                </div>          -->
-<!--            </div>-->
-
-<!--            -->
-<!--            <div class="definition" id="definition-metric-ball">-->
-<!--                <div class="title">metric ball</div>-->
-<!--                <div class="content">  -->
-<!--                    Given \( \epsilon \in \mathbb{R}^{\gt 0} \), consider the set \( B_{d} \left ( x , \epsilon \right ) = \left \lbrace y : d \left ( x , y \right ) < \epsilon \right \rbrace \) of all points \( y \) whose distance from \( x \) is less than \( \epsilon \). It is called the \( \epsilon \)-ball centered at \( x \). Sometimes we omit the metric \( d \) from the notation and write this ball simply as \( B \left ( x , \epsilon \right ) \), when no confusion will arise.    -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            -->
-<!--            <div class="definition" id="definition-toplogy-induced-by-a-metric">-->
-<!--                <div class="title">topology induced by a metric</div>-->
-<!--                <div class="content"> -->
-<!--                    If \( d \) is a <span class="knowledge-link" data-href="#definition-metric-on-a-set">metric</span> on the set \( X \), then the collection of all \( \epsilon \)-balls \( B_{d} \left ( x , \epsilon \right ) \), for \( x \in X \) and \( \epsilon \in \mathbb{R}^{\gt 0} \), is a basis for a topology on \( X \), called the metric topology induced by \( d \)-->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">   -->
-<!--                    If \( X \) is a topological space, \( X \) is said to be metrizable if there exists a metric \( d \) on the set \( X \) that induces the topology of \( X \). A metric space is a metrizable space \( X \) together with a specific metric d that gives the topology of \( X \)   -->
-<!--                </div>          -->
-<!--            </div>-->
-
-<!--            <div class="exercise">-->
-<!--                <div class="title"></div>-->
-<!--                <div class="content">-->
-<!--                    In \( \mathbb{R}^{n} \) define the metric \( d^{'} \left ( x , y \right ) = \left | x_{1} - y_{1} \right |  + \ldots + \left | x_{n} - y_{n} \right |  \) show that \( d^{'} \) is a metric that induces the <span class="knowledge-link" data-href="#definition-standard-topology-on-Rn">standard topology of \( \mathbb{R}^{n} \)</span> -->
-<!--                </div>-->
-<!--            </div>-->
-
-<!--            <div class="proof">-->
-<!--                <div class="title"></div>-->
-<!--                <div class="content">-->
-<!--                    <p>-->
-<!--                        (TODO: show that it's a metric first of all)-->
-<!--                    </p>-->
-<!--                    <p>-->
-<!--                        Since we know that \( d^{'} \) is a metric, then now we can check to see if it induces the usual standard topology of \( \mathbb{R}^{n} \). To show this we'll show that the toplogy induced is equal to the standard toplogy of \( \mathbb{R}^{n} \)-->
-<!--                    </p>-->
-<!--                    <p>-->
-<!--                        One method to show that two topologies are equal is to satisfy the requirements of <span class="knowledge-link" data-href="#corollary-topology-equality-with-basis">topology equality for topologies generated by a basis</span> -->
-<!--                    </p>-->
-<!--                    <p>-->
-<!--                        Let \( p \in \mathbb{R}^{n} \) and let \( p \in B_{d^{'}} \left ( x , \epsilon \right ) \) for some \( x \in \mathbb{R}^{n} \) and \( \epsilon \in \mathbb{R}^{\gt 0} \) be a basis element for the  <span class="knowledge-link" data-href="#definition-toplogy-induced-by-a-metric">topology induced by \( d^{'} \)</span>-->
-<!--                    </p>-->
-<!--                    <p>-->
-<!--                        Recall that \( B_{d^{'}} \left ( x , \epsilon \right ) \) is a <span class="knowledge-link" data-href="#definition-metric-ball">metric ball</span> and thus is equal to \( \left \lbrace y \in \mathbb{R}^{n} : d \left ( x , y \right ) = \left | x_{1} - y_{1} \right |  + \ldots + \left | x_{n} - y_{n} \right |  < \epsilon \right \rbrace \)-->
-<!--                    </p>-->
-<!--                </div>-->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content"> -->
-<!--                    Let X be a metric space with metric d. A subset A of X is said to be-->
-<!--bounded if there is some number M such that-->
-<!--d(a1, a2) ≤ M-->
-<!--for every pair a1, a2 of points of A. If A is bounded and nonempty, the diameter of A-->
-<!--is defined to be the number-->
-<!--diam A = sup{d(a1, a2) | a1, a2 ∈ A}.     -->
-<!--                </div>          -->
-<!--            </div>-->
-<!--            -->
-<!--            <div class="definition">-->
-<!--                <div class="title">                 -->
-<!--                </div>-->
-<!--                <div class="content">  -->
-<!--                    Let X be a metric space with metric d. Define d¯ : X × X → R by-->
-<!--the equation-->
-<!--d¯(x, y) = min{d(x, y), 1}.-->
-<!--Then d¯ is a metric that induces the same topology as d.-->
-<!--The metric d¯ is called the standard bounded metric corresponding to d.    -->
-<!--                </div>          -->
-<!--            </div>-->
         </div>
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