General Topology/Connected spaces

That is, a space is path-connected if and only if between any two points, there is a path.

Here we have a partial converse to the fact that path-connectedness implies connectedness:

This theorem has an important application: It proves that manifolds are connected if and only if they are path-connected. Also, later in this book we'll get to know further classes of spaces that are locally path-connected, such as simplicial and CW complexes.

By substituting "connected" for "path-connected" in the above definition, we get:

Exercises

1. Prove that whenever $X$ is a connected topological space and $Y$ is a topological space and $f:X\to Y$ is a continuous function, then $f(X)$ is connected with the subspace topology induced on it by $Y$.
2. Prove that similarly if $X$ is a path-connected top. space, $Y$ top. space, $f:X\to Y$ continuous, then $f(X)$ is path-connected with the subspace topology induced on it by $Y$.
3. Prove that a topological space $X$ is connected if and only if, when ${\chi }_A:X\to \{0,1\}$ for $A\subseteq X$ denotes the indicator function, the only indicator functions which are continuous are the ones where $A=\mathrm{\varnothing }$ and $A=X$; here $\{0,1\}$ has the discrete topology.
4. Let $X:=\{(0,0)\}\cup \{(x,y)|\mathrm{\exists }n\in \mathbb{N}:y=1/n\}$ with the subspace topology induced by the Euclidean topology of ${ℝ}^2$. Prove that $X$ is not locally connected by proving that $(0,0)$ does not have a connected neighbourhood.