Topology: General Topology 2021

1.

Let \(X\) be a topological space, and let \(x \in X\) be a point. Show that there is a largest connected subspace of \(X\) containing \(x\text{.}\) This is the connected component of \(x\text{.}\)

2.

Let \(X\) be a topological space. Construct a topological space \(P_X\) and a continuous surjection \(\theta \colon X \to P_X\) such that for every \(p \in P_X\text{,}\) the fiber \(\theta^{-1}\{p\}\) is a connected component.

3.

A topological space \(X\) is said to be totally disconnected if and only if the connected component of any \(x \in X \) is the singleton \(\{x\}\text{.}\) Show that the following are equivalent for a topological space \(X\text{:}\)

1. \(X\) is totally disconnected.

2. The map \(\theta \colon X \to P_X\) from the previous exercise is a homeomorphism.

4.

Let \(X\) and \(Y\) be topological spaces, and assume that \(Y\) is totally disconnected. Prove that for any continuous map \(g \colon X \to Y\text{,}\) there exists a unique continuous map \(g' \colon P_X \to Y\) such that \(g = g' \circ \theta\text{,}\) where \(\theta \colon X \to P_X\) is the map constructed above.

5.

A topological space \(X\) is said to be zero-dimensional if and only if every singleton \(\{x\} \subseteq X\) is closed, and there is a base for \(X\) consisting of clopen sets. Show that every zero-dimensional topological space is totally disconnected.

6.

Show that the following topological spaces are zero-dimensional and therefore totally disconnected: any discrete topological space, the rationals \(\QQ \subset \RR\) with the subspace topology, and the Cantor space.