Countability and Compactness

due by Monday, Oct 25, 2021

This assignment is based on material in lectures 16 and 17.

1. Recall that the Sorgenfrey line $\mathbb{R}_l$ is the topological space with underlying set $\mathbb{R}$ and with topology having basis $\{[a, b): a, b\in\mathbb{R}\}$. Let $\{a_n\}$ be a sequence of real numbers. Prove or disprove the following.

1. If $\{a_n\}$ converges to $0$ in $\mathbb{R}$ (with its usual topology), then $\{a_n\}$ converges to $0$ in $\mathbb{R}_l$.
2. If $\{a_n\}$ converges to $0$ in $\mathbb{R}_l$,then $\{a_n\}$ converges to $0$ in $\mathbb{R}$ (with its usual topology).
3. If $a_n=\frac{1}{n}$ for all $n\geq 1$,then $\{a_n\}$ converges to $0$ in $\mathbb{R}_l$.
4. If $a_n=-\frac{1}{n}$ for all $n\geq 1$,then $\{a_n\}$ converges to $0$ in $\mathbb{R}_l$.
2. Let $X$ be the natural numbers with the cofinite topology, and consider the sequence $\{n\}_{n\geq 1}$ in $X$. Prove or disprove the following.

1. The sequence $\{n\}_{n\geq 1}$ converges to $0\in X$.
2. The sequence $\{n\}_{n\geq 1}$ converges to $1\in X$.
3. The sequence $\{n\}_{n\geq 1}$ does not converge in `$X$.
3. Let $S$ be a set. Prove or disprove the following.

1. $S$ with indiscrete topology is compact if and only if $S$ is finite.
2. $S$ with discrete topology is compact if and only if $S$ is finite.
3. $S$ with indiscrete topology is always compact.
4. $S$ with discrete topology is always compact.
4. Let $X$ be an infinite set with the discrete metric. Prove or disprove the following for a subset $A\subset X$.

1. If $A$ is compact then $A$ is closed and bounded.
2. If $A$ is closed and bounded then $A$ is compact.
5. Let $(X, \leq)$ be an ordered set giving a topological space with the order topology. Prove or disprove the following.

1. If $X$ is sequentially compact, then every bounded monotonic sequence in $X$ is convergent.
2. If every bounded monotonic sequence in $X$ is convergent, then $X$ is sequentially compact.