Path connectedness and connectedness

**Problem**

Show that an open set is pathwise connected if and only if it is connected.

**Attempt at Solution**

Suppose is not connected and suppose is pathwise connected. Then, for any points there exists a continuous function which connects and together. Let , where and . Let and . Clearly, we have a contradiction because there cannot exist a continuous function from to if is disconnected. Thus must be connected.

How would I prove the other direction? Is this correct, btw?

Re: Path connectedness and connectedness

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**My Little Pony** **Problem**
Show that an open set

is pathwise connected if and only if it is connected.

**Attempt at Solution**
Suppose

is not connected and suppose

is pathwise connected. Then, for any points

there exists a continuous function

which connects

and

together. Let

, where

and

. Let

and

. Clearly, we have a contradiction because there cannot exist a continuous function from

to

if

is disconnected. Thus

must be connected.

How would I prove the other direction? Is this correct, btw?

More generally, if is a normed vector space then any connected open subset of is path connected.

This follows from the fact that ever locally path connected and connected space is path connected. But, we know that is connected and since every point of has some open ball containing it sitting inside , but since is convex and thus path connected you have that is locally path connected.

If you want to learn more, or see proofs of the above you can look at my blog post here (sorry, the formatting on that one is pretty bad).