- #1

- 394

- 15

## Homework Statement

This is a problem from Munkres(Topology): Show that a connected metric space ##M## having having more than one point is uncountable.

## Homework Equations

A theorem of that section of the book states: Let ##X## be a nonempty compact Hausdorff space. If no singleton in ##X## is open, then ##X## is uncountable

## The Attempt at a Solution

d[/B]If ##M## is connected and has more than one point then no singleton can be an open set since ##\{x\}\,\text{and}\,X\setminus\{x\}## would be a separation. Then if ##M## were compact application of the above mentioned theorem shows that ##M## is uncountable.

However the problem only states that ##M## is connected.

Is it possible that that every connected metric space be compact? or the problem should be solved without using the above mentioned theorem?