### Separation Properties

##### due by Monday, Oct 18, 2021

This assignment is based on material in lecture 15.

1. Let $X$ be a finite set. Prove or disprove the following.

1. If $X$ is Hausdorff then the topology on $X$ is the discrete topology.
2. If $X$ is $T_1$, then the topology on $X$ is the discrete topology.
2. Let $X$ be a countably infinite set with the indiscrete topology. Prove or disprove the following.

1. $X$ is $T_1$.
2. $X$ is Hausdorff.
3. $X$ satisfies the third separation axiom.
4. $X$ satisfies the fourth separation axiom.
3. Let $X$ be a topological space and $f: X \to \mathbb{R}$ be a continuous map. For $c\in \mathbb{R}$ define $X_c = \{x\in X: f(x) < c\}$. Prove or disprove the following.

1. $X_c$ is closed.
2. $X_c$ is open.
3. If $a \leq b$, then $X_a\subset X_b$.
4. If $a < b$, then $\overline{X}_a\subset X_b$.
4. For each of the following spaces $X$, prove or disprove: if $Y$ is a metric space and a sequence of continuous functions $f_n: X\to Y$ converges pointwise to a function $f: X\to Y$, then $f_n$ converges uniformly to $f$.

1. $X$ is finite.
2. $X$ is indiscrete.
3. $X = [0, 1]$.
5. For each of the following subsets $A \subset\mathbb{R}$, prove or disprove: every continuous map $f: A \to [0, 1]$ extends to a continuous map from $\mathbb{R}$ to $[0, 1]$.

1. $A = (0, 1)$.
2. $A = [0, 1]\cup [2, 3]$.
3. $A = \{x\in \mathbb{R}: x \leq 2\}$.