Problem Set - Topology

## Problem Set - Topology

 Define the following and give an example for each: (a)   metric space, (b)   limit of $f$ as $x$ approaches $a$, (c)   limit of $\left({x}_{n}\right)$ as $n\to \infty$, (j)   continuous at $x=a$, (c)   continuous, (d)   uniformly continuous, (e)   Lipschitz, (f)   $\epsilon$-ball, Define the following and give an example for each: (a)   topology, (b)   topological space, (c)   open set, (d)   closed set, (e)   interior, (f)   closure, (g)   interior point, (h)   close point, (i)   neighborhood, (j)   fundamental system of neighborhoods, (k)   continuous at $x=a$, (l)   continuous, Define the following and give an example for each: (a)   topological space, (b)   Hausdorff, (b)   fundamental system of neighborhoods, (b)   basis, (c)   connected set, (d)   compact set, Prove that $ℬ$ and is a basis of $𝒯$ if and only if $ℬ$ satisfies: if $x\in X$ then $ℬ\left(x\right)=\left\{B\in ℬ | x\in B\right\}$ is a fundamental system of neighborhoods of $x$. Let $X$ and $Y$ be metric spaces. Define the topology on $X$ and $Y$. Prove that $f:X\to Y$ is continuous as a function between metric spaces if and only if $f:X\to Y$ is continuous as a function between topological spaces. Define the following and give an example for each: (b)   filter, (c)   finer, (b)   filter base, (b)   neighborhood filter, (d)   limit of $f$ as $x$ approaches $a$, (b)   Fréchet filter, (d)   limit of $\left({x}_{n}\right)$ as $n\to \infty$. Define the following and give an example for each: (c)   ultrafilter, (d)   quasicompact, (d)   Hausdorff, (d)   compact, Let $X$ be a Hausdorff topological space and let $K$ be a compact subset of $X$. Show that $K$ is closed. Let $X$ be a metric space. Show that $X$ is Hausdorff and has a countable basis. Let $X$ be a metric space and let $K$ be a compact subset of $X$. Show that $K$ is closed and bounded. Let $X$ be a metric space and let $E$ be a subset of $X$. Show that $E$ is compact if and only if every infinite subset of $E$ has a limit point in $E$. (What is the definition of limit point???) Let $K$ be a subset of ${ℝ}^{n}$. Show that $K$ is compact if and only if $K$ is closed and bounded. Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous function. Show that if $X$ is connected then $f\left(X\right)$ is connected. Let $E\subseteq ℝ$. Show that $E$ is connected if and only if the set $E$ satisfies if $x,y\in E$ and $z\in ℝ$ and $x then $z\in E$. (Intermediate Value Theorem) Let $f:\left[a,b\right]\to ℝ$ be a continuous function. Show that if $z\in ℝ$ and $f\left(a\right) then there exists $c\in \left(a,b\right)$ such that $f\left(c\right)=z$. Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous function. Show that if $X$ is compact then $f\left(X\right)$ is compact. Let $D$ be a closed bounded subset of $ℝ$ and let $f:D\to ℝ$ be a continuous function. (a)   $f$ is a bounded function, (b)   $f$ attains its maximum and minimum on $D$, (a)   $f$ is uniformly continuous.