The Student Room Group

Subsets

Can the words bounded and closed be used synonymously when talking about subsets??
The same for open and unbounded.
Reply 1
No.
Reply 2
Original post by DFranklin
No.


Are they synonymous geometrically?? And if not may you please explain any difference. (I realise that I did not stipulate 'geometrically' before).

Thanks
Reply 3
[0,)[0, \infty) is an unbounded closed subset of R\mathbb{R}, and (0,1)(0,1) is a bounded open subset... so no.

Or if you want to think more geometrically, consider the plane R2\mathbb{R}^2. Then the x-axis is an unbounded closed set, and the open unit disc {x:x<1}\{ x\, :\, \lVert x \rVert < 1 \} is a bounded open set.
(edited 13 years ago)
Reply 4
Original post by nuodai
[0,)[0, \infty) is an unbounded closed subset of R\mathbb{R}, and (0,1)(0,1) is a bounded open subset... so no.

Or if you want to think more geometrically, consider the plane R2\mathbb{R}^2. Then the x-axis is an unbounded closed set, and the open unit disc {x:x<1}\{ x\, :\, \lVert x \rVert < 1 \} is a bounded open set.


After this post, it is clear I have embarrassed myself with this question :colondollar: Cheers for the answers.
Reply 5
Original post by adie_raz
After this post, it is clear I have embarrassed myself with this question :colondollar: Cheers for the answers.


Not really, it takes a while to get used to coming up with really obvious sounding counter-examples.
Closed does imply bounded in a compact metric space. They're still not synonymous though.

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