Fintie point sets in a Hausdorff space are closed.

[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 U_x and V containing x and x₀ respectively. I then showed that X-{X₀} is the union of all U_x, and it therefore open.

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

 

[micromass]

Yes, your approach works.