Quotient spaces

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.

Let $A_i \cup_{f_i} B_i = A_i \sqcup B_i / x \sim f_i(x), x \in K_i \subset B_i$ be the quotient space of $A_i \cup B_i$ coming from identifying $x \in K_i \subset B_i$ with $f(x) \in A_i$ via a homeomorphism $f_i : K_i \to f(K_i)$, $i = 1, 2$.

Proposition 1.7.6: Suppose there are homeomorphisms $F_A : A_1 \to A_2$, $F_B : B_1 \to B_2$ with $F_B(K_1) = K_2$ and $F_A f_1 = f_2 F_B$. Then the map $F : A_1 \cup_{f_1} B_1 \to A_2 \cup_{f_2} B_2$ given by $F(x) = F_A(x)$ if $x \in A_1$ and $F(x) = F_B(x)$ if $x \in B_1$ is a homeomorphism.

Prove Proposition 1.7.6.

Consider the space $X$ formed from two copies of $R = D^1 \times D^1$ by identifying $\left\{ -1, 1\right\} \times D^1$ to itself via $d$ with $d(1, y) = (1, -y)$ and $d(-1, y) = (-1, -y)$; that is, $X = R \cup_d R$. Construct a homeomorphism between $X$ and the annulus.

Suppose $X = A \cup_g B$, $Y = A \cup_{g'} B$, where $g, g' : K \subset B \to g(K), g'(K) \subset A$ are homeomorphisms. Suppose $(g')^{-1} g : K \to K = h|_K$, where $h : B \to B$ is a homeomorphism. Show that the identity on $A$ and $h$ on $B$ piece together to give a homeomorphism from $X$ to $Y$.

Show that the Möbius band can also be described as a quotient space $D^1 \times D^1 \cup_f D^1 \times D^1$, with $f(-1, t) = (-1, t)$, $f(1, t) = (1, -t)$.

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 $S^2$.