How to visualize compact space?
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I am reading Walter rudin's book on analysis, I am thoroughly enjoying it
although, in the second chapter basic topology i am trying to get a feel for compact spaces, anything that can make that sub chapter more intuitive.
I have been able to come up with imaginative methods for other definitions and theorems, with the help of which i can those concepts , to some extant, understand intuitively.
but compact spaces are a bit tricky to visualize, any help will be useful
Thanks
real-analysis general-topology
asked Nov 14 at 15:59
user199996
1
add a comment |
up vote
0
down vote
favorite
I am reading Walter rudin's book on analysis, I am thoroughly enjoying it
although, in the second chapter basic topology i am trying to get a feel for compact spaces, anything that can make that sub chapter more intuitive.
I have been able to come up with imaginative methods for other definitions and theorems, with the help of which i can those concepts , to some extant, understand intuitively.
but compact spaces are a bit tricky to visualize, any help will be useful
Thanks
real-analysis general-topology
asked Nov 14 at 15:59
user199996
1
2
The Heine-Borel theorem states that a set in Euclidean space $mathbb{R}^n$ is compact if and only if it is closed and bounded. I personally find this latter characterization to be easier to visualize.
– suchan
Nov 14 at 16:00
1
Compact sets are in some sense "small" (hence the name). This is also reflected in the general definition, namely that you should, for any open cover, be able to find a finite subcover.
– MisterRiemann
Nov 14 at 16:04
2
In analysis you can get away with thinking of compactness as sequential compactness, at least up to a point: that is, compact spaces are spaces where sequences have convergent subsequences. This produces a totally opposite intuition to "smallness": it says that compact spaces are spaces that have all the points they "should" have, and in particular correctly suggests that it is possible to make spaces compact by adding points to them ("compactification").
– Qiaochu Yuan
Nov 14 at 20:12
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am reading Walter rudin's book on analysis, I am thoroughly enjoying it
although, in the second chapter basic topology i am trying to get a feel for compact spaces, anything that can make that sub chapter more intuitive.
I have been able to come up with imaginative methods for other definitions and theorems, with the help of which i can those concepts , to some extant, understand intuitively.
but compact spaces are a bit tricky to visualize, any help will be useful
Thanks
real-analysis general-topology
asked Nov 14 at 15:59
user199996
1
I am reading Walter rudin's book on analysis, I am thoroughly enjoying it
although, in the second chapter basic topology i am trying to get a feel for compact spaces, anything that can make that sub chapter more intuitive.
I have been able to come up with imaginative methods for other definitions and theorems, with the help of which i can those concepts , to some extant, understand intuitively.
but compact spaces are a bit tricky to visualize, any help will be useful
Thanks
real-analysis general-topology
real-analysis general-topology
asked Nov 14 at 15:59
user199996
1
asked Nov 14 at 15:59
user199996
1
asked Nov 14 at 15:59
user199996
1
asked Nov 14 at 15:59
user199996
1
asked Nov 14 at 15:59
user199996
1
1
2
The Heine-Borel theorem states that a set in Euclidean space $mathbb{R}^n$ is compact if and only if it is closed and bounded. I personally find this latter characterization to be easier to visualize.
– suchan
Nov 14 at 16:00
1
Compact sets are in some sense "small" (hence the name). This is also reflected in the general definition, namely that you should, for any open cover, be able to find a finite subcover.
– MisterRiemann
Nov 14 at 16:04
2
In analysis you can get away with thinking of compactness as sequential compactness, at least up to a point: that is, compact spaces are spaces where sequences have convergent subsequences. This produces a totally opposite intuition to "smallness": it says that compact spaces are spaces that have all the points they "should" have, and in particular correctly suggests that it is possible to make spaces compact by adding points to them ("compactification").
– Qiaochu Yuan
Nov 14 at 20:12
add a comment |
2
The Heine-Borel theorem states that a set in Euclidean space $mathbb{R}^n$ is compact if and only if it is closed and bounded. I personally find this latter characterization to be easier to visualize.
– suchan
Nov 14 at 16:00
1
Compact sets are in some sense "small" (hence the name). This is also reflected in the general definition, namely that you should, for any open cover, be able to find a finite subcover.
– MisterRiemann
Nov 14 at 16:04
2
In analysis you can get away with thinking of compactness as sequential compactness, at least up to a point: that is, compact spaces are spaces where sequences have convergent subsequences. This produces a totally opposite intuition to "smallness": it says that compact spaces are spaces that have all the points they "should" have, and in particular correctly suggests that it is possible to make spaces compact by adding points to them ("compactification").
– Qiaochu Yuan
Nov 14 at 20:12
2
2
The Heine-Borel theorem states that a set in Euclidean space $mathbb{R}^n$ is compact if and only if it is closed and bounded. I personally find this latter characterization to be easier to visualize.
– suchan
Nov 14 at 16:00
The Heine-Borel theorem states that a set in Euclidean space $mathbb{R}^n$ is compact if and only if it is closed and bounded. I personally find this latter characterization to be easier to visualize.
– suchan
Nov 14 at 16:00
1
1
Compact sets are in some sense "small" (hence the name). This is also reflected in the general definition, namely that you should, for any open cover, be able to find a finite subcover.
– MisterRiemann
Nov 14 at 16:04
Compact sets are in some sense "small" (hence the name). This is also reflected in the general definition, namely that you should, for any open cover, be able to find a finite subcover.
– MisterRiemann
Nov 14 at 16:04
2
2
In analysis you can get away with thinking of compactness as sequential compactness, at least up to a point: that is, compact spaces are spaces where sequences have convergent subsequences. This produces a totally opposite intuition to "smallness": it says that compact spaces are spaces that have all the points they "should" have, and in particular correctly suggests that it is possible to make spaces compact by adding points to them ("compactification").
– Qiaochu Yuan
Nov 14 at 20:12
In analysis you can get away with thinking of compactness as sequential compactness, at least up to a point: that is, compact spaces are spaces where sequences have convergent subsequences. This produces a totally opposite intuition to "smallness": it says that compact spaces are spaces that have all the points they "should" have, and in particular correctly suggests that it is possible to make spaces compact by adding points to them ("compactification").
– Qiaochu Yuan
Nov 14 at 20:12
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The Heine-Borel theorem states that a set in Euclidean space $mathbb{R}^n$ is compact if and only if it is closed and bounded. I personally find this latter characterization to be easier to visualize.
– suchan
Nov 14 at 16:00
1
Compact sets are in some sense "small" (hence the name). This is also reflected in the general definition, namely that you should, for any open cover, be able to find a finite subcover.
– MisterRiemann
Nov 14 at 16:04
2
In analysis you can get away with thinking of compactness as sequential compactness, at least up to a point: that is, compact spaces are spaces where sequences have convergent subsequences. This produces a totally opposite intuition to "smallness": it says that compact spaces are spaces that have all the points they "should" have, and in particular correctly suggests that it is possible to make spaces compact by adding points to them ("compactification").
– Qiaochu Yuan
Nov 14 at 20:12