Separation Properties
due by Monday, Oct 18, 2021
This assignment is based on material in lecture 15.
-
Let
$X$
be a finite set. Prove or disprove the following.- If
$X$
is Hausdorff then the topology on$X$
is the discrete topology. - If
$X$
is$T_1$
, then the topology on$X$
is the discrete topology.
- If
-
Let
$X$
be a countably infinite set with the indiscrete topology. Prove or disprove the following.$X$
is$T_1$
.$X$
is Hausdorff.$X$
satisfies the third separation axiom.$X$
satisfies the fourth separation axiom.
-
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.$X_c$
is closed.$X_c$
is open.- If
$a \leq b$
, then$X_a\subset X_b$
. - If
$a < b$
, then$\overline{X}_a\subset X_b$
.
-
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$f$
.$X$
is finite.$X$
is indiscrete.$X = [0, 1]$
.
-
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$[0, 1]$
.$A = (0, 1)$
.$A = [0, 1]\cup [2, 3]$
.$A = \{x\in \mathbb{R}: x \leq 2\}$
.