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when is a finite topological space path-connected?

Muon

New member
May 15, 2017
2
I'm struggling to understand the intuition behind path-connectedness in finite topological spaces.
Wikipedia https://en.wikipedia.org/wiki/Connected_space#Path_connectedness here claims a finite topological space is connected if and only if it is path-connected. Perhaps this is because connectedness and local path-connectedness imply path-connectedness and finite topological spaces are locally path-connected.
an indiscrete topological space, say $\left\{0,1\right\}$ with the topology $\tau=\left\{\emptyset,\left\{0,1\right\}\right\}$ is path-connected, so there must be a continuous map $f:I\to\left\{0,1\right\}$.
A map $f:X\to Y$ between two topological spaces is continuous if the pre-image of each open set in $Y$ is open in $X$, but what if the path is not a surjective map?
 

GJA

Well-known member
MHB Math Scholar
Jan 16, 2013
253
Hi Muon,

The path $f:[0,1]\rightarrow X$ doesn't need to be surjective (and typically isn't) onto the topological space $X$. Perhaps thinking of testing for path-connectedness a bit more algorithmically might help.

To test whether a space $X$ (finite or infinite) is path-connected:

1) Select two arbitrary points $x$ and $y$ in $X$.

2) Attempt to construct a continuous function $f:[0,1]\rightarrow X$ such that $f(0)=x$ and $f(1) =y$.

3) If step (2) is possible for any choice of $x$ and $y$ in step (1), then $X$ is said to be path-connected. If you are able to select two specific points in step (1) for which it is impossible to complete step (2), then $X$ is not path-connected.

The topology on your (finite) set $X$ will play the central role in determining whether step (2) is possible for any choice of points $x$ and $y$ from step (1).

Moreover, a path-connected space is always connected, so what really needs to be proven is the following: If a finite topological space $X$ is connected, then $X$ is also path connected.