Adjunction spaces and gluing along subsets

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 $f(-1, t) = (-1, t)$, $f(1, t) = (1, -t)$.

Consider the quotient space $X$ formed from two copies of $D^1 \times D^1$ using $f : \left\{ -1, 1\right\} \times D^1 \to \left\{ -1, 1\right\} \times D^1$ by $f(-1, x) = (1, -x)$, $f(1, y) = (-1, y)$, $X = (D^1 \times D^1) \cup_f (D^1 \times D^1)$. Decide whether $X$ is homeomorphic to the annulus or the Möbius band, and prove your assertion.

Show that $D^2 \cup_f D^2$, where $f : K \to K$ is $f(x) = x$, and $K = \left\{ (x_1, x_2) \in S^1 : x_1 \geq 0\right\}$ is homeomorphic to $D^2$.

Construct a homeomorphism between the quotient space $D_{1}^{1} \cup_{\mathrm{id}} D_{2}^{1}$ and the circle $S^{1}$, where $D_{i}^{1}$ denotes a copy of the interval $[-1,1]$ and the identification glues corresponding endpoints of the two copies together via the identity.