Connectedness and Applications
due by Monday, Oct 11, 2021
This assignment is based on material in lectures 12, 13 and 14.

Determine for which of the following topologies on
$\mathbb{R}$
are the corresponding spaces connected. The indiscrete topology.
 The discrete topology.
 The standard topology.
 The cofinite topology.

Let
$X$
be a connected topological space. Prove or disprove the following. If the interior of a set
$A\subset X$
is connected then$A$
is connected.  If
$A\subset X$
is a nonempty proper subset of$X$
(i.e.,$\phi\neq A\neq X$
) then the frontier of$A$
is nonempty.
 If the interior of a set

Let
$f: X \to Y$
be a continuous surjective map. Prove or disprove the following statements. If
$X$
is connected then so is$Y$
.  If
$Y$
is connected then so is$X$
.  The number of connected components of
$Y$
is greater than or equal to the number of connected components of$X$
.  The number of connected components of
$X$
is greater than or equal to the number of connected components of$Y$
.
 If

Let
$X= \{(x, y)\in\mathbb{R}^2: xy = 0\}$
. Then what are the possibilities for the number of components of$X\setminus \{P\}$
for some point$P\in X$
. Prove your answer. 
For which of the following spaces
$X$
is every path in$X$
a constant path.$X = \mathbb{Q}$
with the metric topology from$d(x, y) = x  y$
.$X =\mathbb{N}$
with the cofinite topology.$X = \mathbb{N}$
with the indiscrete topology.