Quotient spaces and identification maps

No proofs, only the correct answers to the question asked, are required for this problem.

If $X$ and $Y$ are topological spaces, the join of $X$ and $Y$ is the quotient space

where

Let $X$ be a normal topological space and $A \subset X$ a closed subspace.

Suppose $q : X \to Y$ and $Y$ has the quotient topology with respect to $(X, q)$ (i.e. $U \subset Y$ is open iff $q^{-1}(U) \subset X$ is open). Show $C \subset Y$ is closed iff $q^{-1}(C) \subset X$ is closed.

Consider $X = [0, 1]$ with $Y$ the quotient space formed by identifying $0$ with $1$, and $X' = \mathbb {R}$ with $Y'$ the quotient space formed by identifying $x$ with $x + n$ for all $n \in \mathbb {Z}$. Let $q : X \to Y$, $q' : X' \to Y'$ be the identification maps, and let $i : X \to X'$ be the inclusion. Since $[i(0)] = [0] = [1] = [i(1)]$, $i$ induces a map $\bar{i} : Y \to Y'$ defined by $\bar{i}([x]) = [i(x)]$, which is a continuous bijection onto $Y'$.

Construct an inverse for $\bar{i}$ and show that it is continuous.

Describe basic open sets in the quotient space $[0, 1] \times [0, 1] / (0, t) \sim (1, t), (s, 0) \sim (s, 1)$ about each of the points $[(0, 0)]$, $[(\tfrac {1}{2}, 0)]$, $[(0, \tfrac {1}{2})]$, and $[(\tfrac {1}{2}, \tfrac {1}{2})]$. Describe the inverse image $q^{-1}(U)$ of each of these basic open sets.

Show that the quotient space formed from a square by identifying all of the points in the bottom edge of the square to each other is homeomorphic to a triangle.

Triangle as a quotient space of the square.

Identify all points in the lower half of the circle to each other. Show that the resulting quotient space $S^1 / \sim$ is homeomorphic to $S^1$.

Put an equivalence relation on the unit disk by making all points on the boundary circle equivalent to each other. Show that the resulting quotient space $D^2 / \sim$ is homeomorphic to $S^2$.

Put an equivalence relation $\sim$ on $\mathbb {R}^{2}$ by saying that two points are equivalent if they both lie on the circle of radius $r$ about the origin. Show that $\mathbb {R}^{2} / \sim$, with the quotient topology, is homeomorphic to $[0, \infty )$.

Identify all points in the lower hemisphere of the sphere $S^2$. Show that the resulting quotient space $S^2 / \sim$ is homeomorphic to $S^2$.

Identify points on the boundary circles of an annulus $A$ between the circles of radius 1 and radius 2 that lie on the same ray from the origin. Show that the resulting quotient space $A / \sim$ is homeomorphic to the torus $T^2$.

Identify the points on the outer circle of an annulus $A$ to one point and the points on the inner circle to a (different) point. Show that $A / \sim$ is homeomorphic to $S^2$. Describe what space you would get if you identified the points on both circles to a single point.

Give a description of the Klein bottle $K$ as a quotient space of $\mathbb {R}^{2}$ analogous to the description of the torus $T$ as a quotient space of $\mathbb {R}^{2}$ given in Chapter 1. Use this description to show that $K$ is a surface.

Show that the map $f: S \to P$, $f(x) = [x]$, arising from considering $P$ (the projective plane) as a quotient space of $S$ (the $2$-sphere) by identifying $x$ with $-x$, has the property that given $y \in P$, there is a neighborhood $U$ of $y$ so that $f^{-1}(U) = U_{1} \sqcup U_{2}$ (a disjoint union) and $f|_{U_{i}}$ is a homeomorphism of $U_{i}$ onto $U$.

Construct a continuous map from the torus $T = S^{1}_{a} \times S^{1}_{b}$ onto the sphere $S$ which sends $S^{1}_{a} \times \left\{ p\right\} \cup \left\{ p\right\} \times S^{1}_{b}$ to the south pole and is a homeomorphism on $T \setminus (S^{1}_{a} \times \left\{ p\right\} \cup \left\{ p\right\} \times S^{1}_{b})$.

Show that the disk with the upper semicircle identified to the lower semicircle as indicated in Figure 2.74(b) is homeomorphic to $P$ (the projective plane).

Show that the disk with identifications on the boundary circle as indicated in Figure 2.74(c) is homeomorphic to $T$ (the torus).

Verify that $T/(z,w) \sim h(z,w)$ is homeomorphic to $K$ where we use our original definition of $K$ as a quotient of $D^1 \times D^1$.

Here $T = S^1 \times S^1$, and $h : T \rightarrow T$ is the homeomorphism given by $h(z,w) = (-z, \bar{w})$, where we regard $z, w$ as complex numbers and $\bar{w}$ is the complex conjugate of $w$.