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One Point Compactification - Why just one?
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I'm reading about one point compactifcation in Munkres. I follow the proof saying that locally compact Hausdorff spaces have one point compactifications. However, I don't understand why the compactification only needs to be one point. Does the same idea work for finitely many added points (or even adding in countably many, or arbitrary amount of points) and assigning the appropriate topology?
general-topology
asked 4 hours ago
yoshi
1,066817
add a comment |Â
up vote
1
down vote
favorite
I'm reading about one point compactifcation in Munkres. I follow the proof saying that locally compact Hausdorff spaces have one point compactifications. However, I don't understand why the compactification only needs to be one point. Does the same idea work for finitely many added points (or even adding in countably many, or arbitrary amount of points) and assigning the appropriate topology?
general-topology
asked 4 hours ago
yoshi
1,066817
1
One point is enough, but you could throw in more. Do you doubt whether one is enough, or did you want to ask about cases where more are added to get a compact space?
– hardmath
4 hours ago
The latter - can you get compactification by adding more than one point? I'm guessing anything is possible with the appropriate topology. But I don't know if there's a more precise answer - I just learned this a couple hours ago.
– yoshi
4 hours ago
1
There are lots of natural compactifications. The two extremes are the 1-pt (minimal) and the Stone-Cech (maximal). But others, like the so-called "end-point compactification" may seem more natural and typically lie somewhere in between. Look 'em up.
– MPW
4 hours ago
Start with the compact unit disk and remove a couple of points from the boundary. Then the original compact disc is a compactification of the resulting space by as many points as you removed.
– Hagen von Eitzen
3 hours ago
Elsewhere in that book, you can read about the Stone-Cech compactification.
– Lord Shark the Unknown
1 hour ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm reading about one point compactifcation in Munkres. I follow the proof saying that locally compact Hausdorff spaces have one point compactifications. However, I don't understand why the compactification only needs to be one point. Does the same idea work for finitely many added points (or even adding in countably many, or arbitrary amount of points) and assigning the appropriate topology?
general-topology
asked 4 hours ago
yoshi
1,066817
I'm reading about one point compactifcation in Munkres. I follow the proof saying that locally compact Hausdorff spaces have one point compactifications. However, I don't understand why the compactification only needs to be one point. Does the same idea work for finitely many added points (or even adding in countably many, or arbitrary amount of points) and assigning the appropriate topology?
general-topology
general-topology
asked 4 hours ago
yoshi
1,066817
asked 4 hours ago
yoshi
1,066817
asked 4 hours ago
yoshi
1,066817
asked 4 hours ago
yoshi
1,066817
asked 4 hours ago
yoshi
1,066817
1,066817
1
One point is enough, but you could throw in more. Do you doubt whether one is enough, or did you want to ask about cases where more are added to get a compact space?
– hardmath
4 hours ago
The latter - can you get compactification by adding more than one point? I'm guessing anything is possible with the appropriate topology. But I don't know if there's a more precise answer - I just learned this a couple hours ago.
– yoshi
4 hours ago
1
There are lots of natural compactifications. The two extremes are the 1-pt (minimal) and the Stone-Cech (maximal). But others, like the so-called "end-point compactification" may seem more natural and typically lie somewhere in between. Look 'em up.
– MPW
4 hours ago
Start with the compact unit disk and remove a couple of points from the boundary. Then the original compact disc is a compactification of the resulting space by as many points as you removed.
– Hagen von Eitzen
3 hours ago
Elsewhere in that book, you can read about the Stone-Cech compactification.
– Lord Shark the Unknown
1 hour ago
add a comment |Â
1
One point is enough, but you could throw in more. Do you doubt whether one is enough, or did you want to ask about cases where more are added to get a compact space?
– hardmath
4 hours ago
The latter - can you get compactification by adding more than one point? I'm guessing anything is possible with the appropriate topology. But I don't know if there's a more precise answer - I just learned this a couple hours ago.
– yoshi
4 hours ago
1
There are lots of natural compactifications. The two extremes are the 1-pt (minimal) and the Stone-Cech (maximal). But others, like the so-called "end-point compactification" may seem more natural and typically lie somewhere in between. Look 'em up.
– MPW
4 hours ago
Start with the compact unit disk and remove a couple of points from the boundary. Then the original compact disc is a compactification of the resulting space by as many points as you removed.
– Hagen von Eitzen
3 hours ago
Elsewhere in that book, you can read about the Stone-Cech compactification.
– Lord Shark the Unknown
1 hour ago
1
1
One point is enough, but you could throw in more. Do you doubt whether one is enough, or did you want to ask about cases where more are added to get a compact space?
– hardmath
4 hours ago
One point is enough, but you could throw in more. Do you doubt whether one is enough, or did you want to ask about cases where more are added to get a compact space?
– hardmath
4 hours ago
The latter - can you get compactification by adding more than one point? I'm guessing anything is possible with the appropriate topology. But I don't know if there's a more precise answer - I just learned this a couple hours ago.
– yoshi
4 hours ago
The latter - can you get compactification by adding more than one point? I'm guessing anything is possible with the appropriate topology. But I don't know if there's a more precise answer - I just learned this a couple hours ago.
– yoshi
4 hours ago
1
1
There are lots of natural compactifications. The two extremes are the 1-pt (minimal) and the Stone-Cech (maximal). But others, like the so-called "end-point compactification" may seem more natural and typically lie somewhere in between. Look 'em up.
– MPW
4 hours ago
There are lots of natural compactifications. The two extremes are the 1-pt (minimal) and the Stone-Cech (maximal). But others, like the so-called "end-point compactification" may seem more natural and typically lie somewhere in between. Look 'em up.
– MPW
4 hours ago
Start with the compact unit disk and remove a couple of points from the boundary. Then the original compact disc is a compactification of the resulting space by as many points as you removed.
– Hagen von Eitzen
3 hours ago
Start with the compact unit disk and remove a couple of points from the boundary. Then the original compact disc is a compactification of the resulting space by as many points as you removed.
– Hagen von Eitzen
3 hours ago
Elsewhere in that book, you can read about the Stone-Cech compactification.
– Lord Shark the Unknown
1 hour ago
Elsewhere in that book, you can read about the Stone-Cech compactification.
– Lord Shark the Unknown
1 hour ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
5
down vote
accepted
It does not need one point. One point suffices to compactify a locally compact space. A compactification can add a lot of extra points (extreme examples are some Cech Stone compactifications) but locally compact spaces are special because they alone have a "minimal compactification". Having only one point in this so-called "remainder" (the "extra point(s)" we have added) is often handy (we can easily extend some continuous functions from $X$ to its one-point compactification e.g., which is used in functional analysis etc) and intuitive (Riemann sphere for the complex plane e.g.). If a space has a compactification with a closed remainder (e.g. a finite one) then $X$ must have been a locally compact (as $X$ is then homeomorphic to an open subset of a compact (Hausdorff) space to start with.
answered 4 hours ago
Henno Brandsma
95.8k343104
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
It does not need one point. One point suffices to compactify a locally compact space. A compactification can add a lot of extra points (extreme examples are some Cech Stone compactifications) but locally compact spaces are special because they alone have a "minimal compactification". Having only one point in this so-called "remainder" (the "extra point(s)" we have added) is often handy (we can easily extend some continuous functions from $X$ to its one-point compactification e.g., which is used in functional analysis etc) and intuitive (Riemann sphere for the complex plane e.g.). If a space has a compactification with a closed remainder (e.g. a finite one) then $X$ must have been a locally compact (as $X$ is then homeomorphic to an open subset of a compact (Hausdorff) space to start with.
answered 4 hours ago
Henno Brandsma
95.8k343104
add a comment |Â
up vote
5
down vote
accepted
It does not need one point. One point suffices to compactify a locally compact space. A compactification can add a lot of extra points (extreme examples are some Cech Stone compactifications) but locally compact spaces are special because they alone have a "minimal compactification". Having only one point in this so-called "remainder" (the "extra point(s)" we have added) is often handy (we can easily extend some continuous functions from $X$ to its one-point compactification e.g., which is used in functional analysis etc) and intuitive (Riemann sphere for the complex plane e.g.). If a space has a compactification with a closed remainder (e.g. a finite one) then $X$ must have been a locally compact (as $X$ is then homeomorphic to an open subset of a compact (Hausdorff) space to start with.
answered 4 hours ago
Henno Brandsma
95.8k343104
add a comment |Â
up vote
5
down vote
accepted
up vote
5
down vote
accepted
It does not need one point. One point suffices to compactify a locally compact space. A compactification can add a lot of extra points (extreme examples are some Cech Stone compactifications) but locally compact spaces are special because they alone have a "minimal compactification". Having only one point in this so-called "remainder" (the "extra point(s)" we have added) is often handy (we can easily extend some continuous functions from $X$ to its one-point compactification e.g., which is used in functional analysis etc) and intuitive (Riemann sphere for the complex plane e.g.). If a space has a compactification with a closed remainder (e.g. a finite one) then $X$ must have been a locally compact (as $X$ is then homeomorphic to an open subset of a compact (Hausdorff) space to start with.
answered 4 hours ago
Henno Brandsma
95.8k343104
It does not need one point. One point suffices to compactify a locally compact space. A compactification can add a lot of extra points (extreme examples are some Cech Stone compactifications) but locally compact spaces are special because they alone have a "minimal compactification". Having only one point in this so-called "remainder" (the "extra point(s)" we have added) is often handy (we can easily extend some continuous functions from $X$ to its one-point compactification e.g., which is used in functional analysis etc) and intuitive (Riemann sphere for the complex plane e.g.). If a space has a compactification with a closed remainder (e.g. a finite one) then $X$ must have been a locally compact (as $X$ is then homeomorphic to an open subset of a compact (Hausdorff) space to start with.
answered 4 hours ago
Henno Brandsma
95.8k343104
answered 4 hours ago
Henno Brandsma
95.8k343104
answered 4 hours ago
Henno Brandsma
95.8k343104
answered 4 hours ago
Henno Brandsma
95.8k343104
95.8k343104
add a comment |Â
add a comment |Â
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1
One point is enough, but you could throw in more. Do you doubt whether one is enough, or did you want to ask about cases where more are added to get a compact space?
– hardmath
4 hours ago
The latter - can you get compactification by adding more than one point? I'm guessing anything is possible with the appropriate topology. But I don't know if there's a more precise answer - I just learned this a couple hours ago.
– yoshi
4 hours ago
1
There are lots of natural compactifications. The two extremes are the 1-pt (minimal) and the Stone-Cech (maximal). But others, like the so-called "end-point compactification" may seem more natural and typically lie somewhere in between. Look 'em up.
– MPW
4 hours ago
Start with the compact unit disk and remove a couple of points from the boundary. Then the original compact disc is a compactification of the resulting space by as many points as you removed.
– Hagen von Eitzen
3 hours ago
Elsewhere in that book, you can read about the Stone-Cech compactification.
– Lord Shark the Unknown
1 hour ago