Is connected component open?

Question

There is a theorem that:A space is locally connected iff each connected components of an open set is open.

But recently I had seen to prove That each connected component is closed. Connected Components are Closed

Then how can the connected component of an open set be open if it is a locally connected space ? It will be contradiction to the statement that connected components is closed.

Think about $\mathbb{R} \setminus {0}$. What are the connected components? — Carl Mummert, Mar 06 '19 at 02:12
In fact, it is true in general that any connected component is clopen (it is both open and closed). If you want a proof, you should add your definition of a "connected component" first. — stressed out, Mar 06 '19 at 13:41
sets are not doors, they can be open and closed at the same time... — Henno Brandsma, Mar 06 '19 at 17:36

Kaj Hansen · Accepted Answer · 2019-03-23T22:55:04.877

4

A subset being closed doesn't preclude that subset from being open. For a simple example, every discrete space is locally connected, and every subset of a discrete space—in particular the singleton sets (which are the connected components)—is both open and closed.

edited Mar 23 '19 at 22:55

answered Mar 06 '19 at 02:03

Kaj Hansen

33,011

Is connected component open?

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