Topological Quotients Review

# Topological Quotients Review

We will now review some of the recent material regarding topological quotients.

• Recall from the Topological Quotients page that if $(X, \tau)$ is a topological space and $\sim$ is an equivalence relation on $X$ with $X / \sim$ denoting the set of all equivalence classes of $X$ then the canonical Quotient Map $q : X \to X \: / \sim$ is defined for all $x \in X$ by $q(x) = [x]$. In other words, each element in $X$ is sent to its equivalence class in $X \: / \sim$.
• Furthermore, if $(X, \tau)$ is a topological space and $\sim$ is an equivalence relation on $X$ then the Quotient Topology on $X \: / \sim$ is defined to be the final topology on $X \: / \sim$ induced by the canonical quotient map $q$ defined above. The resulting Topological Quotient is the set of equivalence classes $X / \sim$ with the quotient topology.
• In other words, the topological quotient $X / \sim$ can be constructed as follows. Take a set $U$ of equivalence classes in $X \: / \sim$. Then consider the inverse image $q^{-1}(U)$ in $X$. If this set is in $X$ then declare $U$ to be open in $X \: / \sim$. The collection of all such sets is the quotient topology on $X \: / \sim$.
• We then looked at when sets are open/closed in the topological quotient $X \: / \sim$ on the Open and Closed Sets in Topological Quotients page. We saw that a set $U$ is open in $X \: / \sim$ if and only $\displaystyle{\bigcup_{[x] \in U} [x]}$ is open in $X$. Intuitively, this should be clear (try to make sense of this with the diagram of the quotient map given above).
• Similarly, we saw that a set $C$ is closed in $X \: / \sim$ if and only if $\displaystyle{\bigcup_{[x] \in C} [x]}$ is closed in $X$.
• On the Topological Quotients in Euclidean Space page we looked at some examples of topological quotients in Euclidean space. We joined ends of a closed interval $[0, 1]$ to form a circle. We also constructed a hollow torus from the closed unit square $[0, 1] \times [0, 1]$.
• On the Topological Sums of Topological Spaces page we looked at constructing a new type of topological space. We saw that if $\{ (X_i, \tau_i) : i \in I \}$ is an arbitrary collection of pairwise disjoint topological spaces then we defined the Topological Sum denoted $\displaystyle{\bigoplus_{i \in I} X_i}$ to be the topological space of the set $\displaystyle{\bigcup_{i \in I} X_i}$ and whose topology $\tau$ is generated by the basis:
(1)
\begin{align} \quad \mathcal B = \left \{ U \subseteq \bigcup_{i \in I} X_i : U \in \tau_i \: \mathrm{for \: some \:} i \in I \right \} \end{align}
• In other words, if $\{ (X_i, \tau_i) : i \in I \}$ is a collection of pairwise disjoint topological spaces then an open set in the topological sum $\displaystyle{\bigoplus_{i \in I} X_i}$ is a union of sets who can be distinguished to be open in the summed spaces $X_i$.
• We then looked at a very important lemma on The Gluing Lemma page call the Gluing Lemma. It states that if $X$ and $Y$ are topological spaces, $A, B \subseteq X$, $X = A \cup B$, $f : A \to Y$, $g : B \to Y$, and $f(x) = g(x)$ for all $x \in A \cap B$ then the "glued" map $h : X \to Y$ defined by $\left\{\begin{matrix} f(x) & \mathrm{if} \: x \in A \\ g(x) & \mathrm{if} \: x \in B \end{matrix}\right.$ is continuous.
• On the Gluing Topological Spaces to Themselves page we looked at… well… gluing topological spaces to themselves! We saw that if $Z$ is a topological space, $X, Y \subseteq Z$ are disjoint, and $f : X \to Y$, then we can define an equivalence relation $\sim$ on $Z$ where we say that $a \sim b$ if $f(a) = b$ and also $x \sim x$ for all $x \in Z$. Then the equivalence classes of $Z$ are $\{ \{ y \} \cup f^{-1}(y) \}$ for $y \in Z$. The resulting space is denoted $Z_f$ and is called the Gluing of $X$ and $Y$ along $f$.
• With this notion, we the circular faces to a cylinder together to (once again) obtain a torus.
• On the Gluing Disjoint Topological Spaces page we looked at gluing disjoint topological spaces together. If $X$ and $Y$ are disjoint topological spaces and if $f : A \to Y$ is a map then we can define an equivalence relation $\sim$ on $X \cup Y$ by saying that $a \sim b$ if $f(a) = b$ and $x \sim x$ for all $x \in X \cup Y$. Like above, the equivalence classes of $X \cup Y$ are $\{ \{ y \} \cup f^{-1}(y) \}$ for $y \in X \cup Y$. The resulting space is denoted $(X \oplus Y) \: / \sim$ or $X \cup_f Y$ and is called the Gluing of $X$ and $Y$ along $f$.
• With this notion, we took two disjoint cylinders and glued a circle face of one of the cylinders to a circular face of the other cylinder.
• On the Construction of the Möbius Strip page we constructed the mobius strip. We took the unit square $[0, 1] \times [0, 1]$ and looked the vertical sides which we denoted as subsets $X$ and $Y$. We then defined a function $f : X \to Y$ by $f(0, x) = (1, 1 - x)$. By gluing $X$ and $Y$ along $f$ we obtained the Möbius strip.