Why it is needed to show $X$ is a subspace of $Y$ in the one point compactification theorem?
$begingroup$
Why it is needed to show $X$ is a subspace of $Y$ in the one point compactification theorem?
The Theorem is the following.
" X be a space. Then $X$ is locally compact Hausdorff iff there exists a space $Y$ satisfying the following conditions
1) $X$ is a subspace of $Y$.
2) The set $Y-X$ consists of single point.
3) $Y$ is a compact Hausdorff space. "
$X$ is a subset of $Y$ . Then why $X$ is not becoming naturally a subspace of $Y$?
Can anyone please make me understand ?
general-topology compactness
edited Dec 30 '18 at 16:23
Henno Brandsma
117k349127
asked Dec 30 '18 at 15:30
cmicmi
1,159313
$endgroup$
add a comment |
$begingroup$
Why it is needed to show $X$ is a subspace of $Y$ in the one point compactification theorem?
The Theorem is the following.
" X be a space. Then $X$ is locally compact Hausdorff iff there exists a space $Y$ satisfying the following conditions
1) $X$ is a subspace of $Y$.
2) The set $Y-X$ consists of single point.
3) $Y$ is a compact Hausdorff space. "
$X$ is a subset of $Y$ . Then why $X$ is not becoming naturally a subspace of $Y$?
Can anyone please make me understand ?
general-topology compactness
edited Dec 30 '18 at 16:23
Henno Brandsma
117k349127
asked Dec 30 '18 at 15:30
cmicmi
1,159313
$endgroup$
add a comment |
$begingroup$
Why it is needed to show $X$ is a subspace of $Y$ in the one point compactification theorem?
The Theorem is the following.
" X be a space. Then $X$ is locally compact Hausdorff iff there exists a space $Y$ satisfying the following conditions
1) $X$ is a subspace of $Y$.
2) The set $Y-X$ consists of single point.
3) $Y$ is a compact Hausdorff space. "
$X$ is a subset of $Y$ . Then why $X$ is not becoming naturally a subspace of $Y$?
Can anyone please make me understand ?
general-topology compactness
edited Dec 30 '18 at 16:23
Henno Brandsma
117k349127
asked Dec 30 '18 at 15:30
cmicmi
1,159313
$endgroup$
Why it is needed to show $X$ is a subspace of $Y$ in the one point compactification theorem?
The Theorem is the following.
" X be a space. Then $X$ is locally compact Hausdorff iff there exists a space $Y$ satisfying the following conditions
1) $X$ is a subspace of $Y$.
2) The set $Y-X$ consists of single point.
3) $Y$ is a compact Hausdorff space. "
$X$ is a subset of $Y$ . Then why $X$ is not becoming naturally a subspace of $Y$?
Can anyone please make me understand ?
general-topology compactness
general-topology compactness
edited Dec 30 '18 at 16:23
Henno Brandsma
117k349127
asked Dec 30 '18 at 15:30
cmicmi
1,159313
edited Dec 30 '18 at 16:23
Henno Brandsma
117k349127
asked Dec 30 '18 at 15:30
cmicmi
1,159313
edited Dec 30 '18 at 16:23
Henno Brandsma
117k349127
edited Dec 30 '18 at 16:23
Henno Brandsma
117k349127
edited Dec 30 '18 at 16:23
Henno Brandsma
117k349127
117k349127
asked Dec 30 '18 at 15:30
cmicmi
1,159313
asked Dec 30 '18 at 15:30
cmicmi
1,159313
asked Dec 30 '18 at 15:30
cmicmi
1,159313
1,159313
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You start with a space $mathbf X=(X,tau_X)$ where $X$ denotes the underlying set and $tau_X$ denotes a topology on this set.
Now a locally compact Hausdorff space $mathbf Y=(Y,tau_Y)$ might exist such that $Xsubseteq Y$ and $Y-X$ is a singleton, but that on its own says at most that $Y$ is compactification of space $(X,rho)$ where $rho$ denotes the subspace topology on $X$ inherited from $mathbf Y$.
So it says nothing yet about our original space $mathbf X=(X,tau_X)$.
To make that change it must be demanded that $tau_X$ and $rho$ coincide or equivalently that $mathbf X=(X,tau_X)$ is a subspace of $mathbf Y$.
answered Dec 30 '18 at 15:55
drhabdrhab
104k545136
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
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oldest
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active
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votes
$begingroup$
You start with a space $mathbf X=(X,tau_X)$ where $X$ denotes the underlying set and $tau_X$ denotes a topology on this set.
Now a locally compact Hausdorff space $mathbf Y=(Y,tau_Y)$ might exist such that $Xsubseteq Y$ and $Y-X$ is a singleton, but that on its own says at most that $Y$ is compactification of space $(X,rho)$ where $rho$ denotes the subspace topology on $X$ inherited from $mathbf Y$.
So it says nothing yet about our original space $mathbf X=(X,tau_X)$.
To make that change it must be demanded that $tau_X$ and $rho$ coincide or equivalently that $mathbf X=(X,tau_X)$ is a subspace of $mathbf Y$.
answered Dec 30 '18 at 15:55
drhabdrhab
104k545136
$endgroup$
add a comment |
$begingroup$
You start with a space $mathbf X=(X,tau_X)$ where $X$ denotes the underlying set and $tau_X$ denotes a topology on this set.
Now a locally compact Hausdorff space $mathbf Y=(Y,tau_Y)$ might exist such that $Xsubseteq Y$ and $Y-X$ is a singleton, but that on its own says at most that $Y$ is compactification of space $(X,rho)$ where $rho$ denotes the subspace topology on $X$ inherited from $mathbf Y$.
So it says nothing yet about our original space $mathbf X=(X,tau_X)$.
To make that change it must be demanded that $tau_X$ and $rho$ coincide or equivalently that $mathbf X=(X,tau_X)$ is a subspace of $mathbf Y$.
answered Dec 30 '18 at 15:55
drhabdrhab
104k545136
$endgroup$
add a comment |
$begingroup$
You start with a space $mathbf X=(X,tau_X)$ where $X$ denotes the underlying set and $tau_X$ denotes a topology on this set.
Now a locally compact Hausdorff space $mathbf Y=(Y,tau_Y)$ might exist such that $Xsubseteq Y$ and $Y-X$ is a singleton, but that on its own says at most that $Y$ is compactification of space $(X,rho)$ where $rho$ denotes the subspace topology on $X$ inherited from $mathbf Y$.
So it says nothing yet about our original space $mathbf X=(X,tau_X)$.
To make that change it must be demanded that $tau_X$ and $rho$ coincide or equivalently that $mathbf X=(X,tau_X)$ is a subspace of $mathbf Y$.
answered Dec 30 '18 at 15:55
drhabdrhab
104k545136
$endgroup$
You start with a space $mathbf X=(X,tau_X)$ where $X$ denotes the underlying set and $tau_X$ denotes a topology on this set.
Now a locally compact Hausdorff space $mathbf Y=(Y,tau_Y)$ might exist such that $Xsubseteq Y$ and $Y-X$ is a singleton, but that on its own says at most that $Y$ is compactification of space $(X,rho)$ where $rho$ denotes the subspace topology on $X$ inherited from $mathbf Y$.
So it says nothing yet about our original space $mathbf X=(X,tau_X)$.
To make that change it must be demanded that $tau_X$ and $rho$ coincide or equivalently that $mathbf X=(X,tau_X)$ is a subspace of $mathbf Y$.
answered Dec 30 '18 at 15:55
drhabdrhab
104k545136
edited Dec 30 '18 at 16:03
answered Dec 30 '18 at 15:55
drhabdrhab
104k545136
answered Dec 30 '18 at 15:55
drhabdrhab
104k545136
answered Dec 30 '18 at 15:55
drhabdrhab
104k545136
104k545136
add a comment |
add a comment |
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