Open connected subsets of path connected spaces

Question

Does every open and connected subset of path connected topological space has to be path connected? Statement should be false as there is a similar theorem but for Euclidean spaces, however I can't think of a counterexample. What about the same statement, but for path connected metric spaces?

Answer 1

Consider the space $$X=\left\{(x,y)\in\Bbb R^2\,:\, (x=0\land y\le 2)\lor \left(x\ne 0\land y=\sin\frac1{\lvert x\rvert}\right) \lor (y\ge 0\land x^2+y^2=4)\right\}$$

I.e. a topologist sine, plus an appropriate vertical half-line, plus a half circle "path-connecting" the curve to the tip of the half-line. Then, $X\setminus \{(0,2)\}$ is connected, but not path-connected.

Answer 2

The classical example of a connected (metric) space that is not path-connected is the topologist's sine curve. I will give an example based on this.

Consider the graph of the $\sin(\frac{1}{x})$ function on $(0,1]$.

Let $X$ be the space which consists of this graph together with the vertical line segment connecting $(0,-1)$ and $(0,1)$, and the curve in red:

$X$ is a metric space that is path connected. You can also clearly see this as an open subset of $X$:

This is an open connected subspace of the $X$ that is not path connected.

This counter example is a metric space. It applies as well for the general case of topological spaces.