More Topologies and Homeomorphisms
due by Monday, Sep 20, 2021
This assignment is based on material in lectures 9, 10 and 11.

Let
$X$
be a set. What is the initial topology on$X$
so that all functions$f: X \to \{0, 1\}$
are continuous? Prove your result. 
Prove or disprove the following.
 Every isometric embedding is continuous.
 The composition of isometric embeddings is an isometric embedding.

Prove or disprove that there exists a continuous bijection from
$X$
to$Y$
for the following pairs of spaces.$X=(0, 1]$ and $Y= S^1$
.$X = (0, 1)\cup [2, 3)$ and $Y= (1, 2)$
.

Prove or disprove that the following spaces are homeomorphic to
$\mathbb{R^2}$
.$\{(x, y)\in \mathbb{R^2}: x^2 + 2y^2 < 1 \}$
.$\{(x, y)\in \mathbb{R^2}: x^2  2y^2 < 1 \}$
.

Prove that the order topology on a finite set
$S$
with respect to any order is the discrete topology on$S$
. 
Given spaces
$X$
and associated covers$\Gamma$
, prove or disprove in each case that$\Gamma$
is a fundamental cover of $X$.$X = [0, 2]$
and$\Gamma = \{[0, 1], [1, 2]\}$
.$X = [0, 2]$
and$\Gamma = \{[0, 1], (1, 2]\}$
.$X =\mathbb{R}$
and$\Gamma=\{\mathbb{Q}, \mathbb{R}\setminus \mathbb{Q}\}$
.