Countability and Compactness
due by Monday, Oct 25, 2021
This assignment is based on material in lectures 16 and 17.

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. If
$\{a_n\}$
converges to$0$
in$\mathbb{R}$
(with its usual topology), then$\{a_n\}$
converges to$0$
in `$\mathbb{R}_l$.  If
$\{a_n\}$
converges to$0$
in$\mathbb{R}_l$
,then$\{a_n\}$
converges to$0$
in$\mathbb{R}$
(with its usual topology).  If
$a_n=\frac{1}{n}$
for all$n\geq 1$
,then$\{a_n\}$
converges to$0$
in `$\mathbb{R}_l$.  If
$a_n=\frac{1}{n}$
for all$n\geq 1$
,then$\{a_n\}$
converges to$0$
in `$\mathbb{R}_l$.
 If

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. The sequence
$\{n\}_{n\geq 1}$
converges to `$0\in X$.  The sequence
$\{n\}_{n\geq 1}$
converges to `$1\in X$.  The sequence
$\{n\}_{n\geq 1}$
does not converge in `$X$.
 The sequence

Let
$S$
be a set. Prove or disprove the following.$S$
with indiscrete topology is compact if and only if$S$
is finite.$S$
with discrete topology is compact if and only if$S$
is finite.$S$
with indiscrete topology is always compact.$S$
with discrete topology is always compact.

Let
$X$
be an infinite set with the discrete metric. Prove or disprove the following for a subset$A\subset X$
. If
$A$
is compact then$A$
is closed and bounded.  If
$A$
is closed and bounded then$A$
is compact.
 If

Let
$(X, \leq)$
be an ordered set giving a topological space with the order topology. Prove or disprove the following. If
$X$
is sequentially compact, then every bounded monotonic sequence in$X$
is convergent.  If every bounded monotonic sequence in
$X$
is convergent, then$X$
is sequentially compact.
 If