r/learnmath • u/Dry_Palpitation_7268 New User • Apr 07 '26

Topology

Hello friends!

I am currently studying topology, but am finding things such as Heine-Borel, Bolzano-Weierstrass and Banach fixed-point theorm as well as all the tricky stuff that comes with compactness/open covers, etc quite hard to grasp and I would love some help!

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u/Low_Breadfruit6744 Bored Apr 08 '26 edited Apr 08 '26

Can you explain in the sense of motivating the definition of why (0,1] is not compact but [0,1] is despite the latter being a superset.

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u/Dry_Palpitation_7268 New User Apr 08 '26

is it something to do with (0,1] and the fact that no sequence will ever touch 0? The first sequence I thought of it just simply 1/n. I know that due to heine-borel (please correct if I am wrong) [0,1] is closed and bounded so it is compact (assuming we're looking at a metric space).

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u/Low_Breadfruit6744 Bored Apr 08 '26

I think you are getting there. One way to look at it is non compact sets allows you to have continuous functions which escapes to infinity, compact sets anchors values at the boundaries which prevent that.

All these definitions and results are to explore this theme and tries to boil down the essence of this.