# Homework Help: Fintie point sets in a Hausdorff space are closed.

1. Nov 10, 2012

### jmjlt88

This may seem like a silly question, but I'll ask it anyways. :)

In the Munkres text, he proves this by showing that one-point sets are closed, which I completely understand why it follows that finite point sets are closed. He does so by showing that the arbitrary one-point set {x0} equals its own closure, and therefore closed (which, again I completely understand). Now, before I read a proof, I like to see try it myself. When I did it, I also proved that one-point sets are closed, but I did so by showing X-{x0} is open. I did this by noting that for each x in X-{x0}, there is disjoint neighborhoods Ux and V containing x and x0 respectively. I then showed that X-{X0} is the union of all Ux, and it therefore open.

My question is, does this approach work as well? Thank you!

2. Nov 10, 2012