jmjlt88
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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!
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!