Connectedness and Applications

due by Monday, Oct 11, 2021

This assignment is based on material in lectures 12, 13 and 14.

1. Determine for which of the following topologies on $\mathbb{R}$ are the corresponding spaces connected.

   1. The indiscrete topology.
   2. The discrete topology.
   3. The standard topology.
   4. The cofinite topology.

2. Let $X$ be a connected topological space. Prove or disprove the following.

   1. If the interior of a set $A\subset X$ is connected then $A$ is connected.
   2. If $A\subset X$ is a non-empty proper subset of $X$ (i.e., $\phi\neq A\neq X$) then the frontier of $A$ is non-empty.

3. Let $f: X \to Y$ be a continuous surjective map. Prove or disprove the following statements.

   1. If $X$ is connected then so is $Y$.
   2. If $Y$ is connected then so is $X$.
   3. The number of connected components of $Y$ is greater than or equal to the number of connected components of $X$.
   4. The number of connected components of $X$ is greater than or equal to the number of connected components of $Y$.

4. 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.

5. For which of the following spaces $X$ is every path in $X$ a constant path.

   1. $X = \mathbb{Q}$ with the metric topology from $d(x, y) = |x - y|$.
   2. $X =\mathbb{N}$ with the cofinite topology.
   3. $X = \mathbb{N}$ with the indiscrete topology.