Topologies and topological spaces, continuous maps and homeomorphisms, subspaces, product spaces, quotient spaces, compactness and connectedness.

这是一份Bath巴斯大学MA30055作业代写的成功案

$D_{p} \subset U_{j}$ with $p \in D_{p}\left(D_{p}\right.$ is the union of ${p}$ with a disc tangent at $\left.p\right)$. $\bigcup_{p \in F_{1}} D_{p} \subset U_{1}$ and $\bigcup_{p \in F_{2}} D_{p} \subset U_{2}$ are also disjoint open sets containing $F_{1}$ and $F_{2}$ respectively. Let $f: R^{1} \rightarrow R^{1}$ be defined by $f(p)=$ radius of $D_{p} .(f$ is not necessarily continuous. $)$ For each positive integer $n$, let
$$
T_{n}=\left[p \in F_{2} ; f(p) \geq \frac{1}{n}\right] .
$$
Then $\bigcup_{n=1}^{\infty} T_{n}=F_{2}$ because $p \in F_{2}$ implies $f(p) \geq 1 / n$ for some $n$. Moreover,
$$
R^{1}=\left(\bigcup_{n=1}^{\infty} T_{n}\right) \cup\left(\bigcup_{q \in F_{1}}{q}\right),
$$

MA30055 COURSE NOTES ：

(1) Let $F: X \times I \rightarrow Y$ be defined by $F(x, t)=f(x) . \quad F$ is continuous because it is the composition of the continuous maps $f$ and projection onto the first factor.
(2) Given a homotopy $F: X \times I \rightarrow Y$ such that $F(x, 0)=f(x)$ and $F(x, 1)=g(x)$, let $G: X \times I \rightarrow Y$ be defined by $G(x, t)=F(x, 1-t)$. $G$ is continuous because $t \rightarrow 1-t$ is continuous. $G$ is a homotopy from $g$ to $f$.
(3) Given homotopies $F, G: X \times I \rightarrow Y$, with $F(x, 0)=f(x), F(x, 1)=g(x)=G(x, 0)$, and $G(x, 1)=h(x)$, let $H: X \times I \rightarrow Y$ be defined by
$$
H(x, t)= \begin{cases}F(x, 2 t) & (0 \leq t \leq 1 / 2) \ G(x, 2 t-1) & (1 / 2 \leq t \leq 1)\end{cases}
$$