# Prove the 1 point compactification of a Hausdorff space X is a compact space which contains X as a dense subspace

#### Cratylus

##### Junior Member
I am using Alexandroff 1-point compactification. I am clueless how to prove it

Here is a summary of the definition:
Let X [imath]\cup[/imath]{p} with topology defined by 1) nbhds of points of X are the same as in the
topology on X and 2) U [imath]\subset[/imath] p[imath]\notin[/imath] X is a basic open nbhd of p iff p[imath]\in[/imath] U and p[imath]\notin[/imath] X -U.

#### blamocur

##### Senior Member
Where does your definition come from? In particular the 2) clause does not make sense: [imath]p\in U[/imath] and [imath]p\notin X -U[/imath] ??
I found the definition in Wikipedia (https://en.wikipedia.org/wiki/Alexandroff_extension) to be pretty clear.

#### Cratylus

##### Junior Member
My text. The text uses a symbol for Alexandroff compactication that I cannot duplicate. nbhd is neighborhood. iff is if and only if

#### blamocur

##### Senior Member
The exact symbol does not matter, but you still haven't addressed my question about about "p in U and p not in U", which is always false.

What do you find difficult to prove? That [imath]X\cup {p}[/imath] is compact? That [imath]X[/imath] is dense in [imath]X\cup {p}[/imath]? Both?

Which definition of compact space are you using?

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• topsquark

#### Cratylus

##### Junior Member
I tried Latex formatting, but it was messy,so I gave a pic

Let p be an object not in X and [imath]\tilde{ X}[/imath] be the set X [imath]\cup[/imath] {p} with a Hausdorff space..

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#### blamocur

##### Senior Member
Here is the beginning of a proof that [imath]\tilde X[/imath] is compact:

Let [imath]\{U_s\}[/imath] be a collection of open sets covering [imath]\tilde X[/imath], where [imath]s\in S[/imath] and [imath]\bigcup_{s\in S} U_s = \tilde X[/imath]. We need to show that there is a finite subset [imath]T \subset S[/imath] such that [imath]\bigcup_{t\in T} U_t = \tilde X[/imath].

If we take an arbitrary [imath]U_0[/imath] from [imath]\{U_s\}[/imath] such that [imath]p\in U_0[/imath] then what can we say about [imath]\tilde X - U_0[/imath] ?

#### Cratylus

##### Junior Member
X-U0 is compact. If it is compact in T2 it is closed. Then by using closed Def of sub space F=A \cup X-U0

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#### blamocur

##### Senior Member
X-U0 is compact. If it is compact in T2 it is closed. Then by using closed Def of sub space F=A \cup X-U0
What do you mean? What is [imath]A[/imath]? Can you prove that [imath]\tilde X[/imath] is compact?

#### Cratylus

##### Junior Member
What do you mean? What is [imath]A[/imath]? Can you prove that [imath]\tilde X[/im [/QUOTE][/imath]

Since U0 is compact X is indeed an open subset of one-point compactification

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#### blamocur

##### Senior Member
Since U0 is compact X is indeed an open subset of one-point compactification

Why do you think that [imath]U_0[/imath] is compact?

#### blamocur

##### Senior Member
The answer there says a set is open if it is a subset of [imath]X[/imath] and is open, or (more relevantly to my question) if it is of the form [imath]{p}\cup U[/imath], where [imath]U\in x[/imath] has compact complement (I've replaced [imath]\infty[/imath] with [imath]p[/imath]).

#### Cratylus

##### Junior Member
So [imath]U_0[/imath] isn’t the compact complement. I thought it was.
See page 115

#### blamocur

##### Senior Member
So [imath]U_0[/imath] isn’t the compact complement. I thought it was.
See page 115

What does it mean to be "the compact complement"?

Also, do you want to try answering my post #8?

#### Cratylus

##### Junior Member
You stated
“where U∈X has compact complement”

#### blamocur

##### Senior Member
You stated
“where U∈X has compact complement”

Which is different from "is" or "isn't" compact complement.
You still haven't answered my question in #8.

• topsquark

#### Cratylus

##### Junior Member
I guess [imath]X-U_0[/imath] is not compact,or I don’t know the answer

#### blamocur

##### Senior Member
I guess [imath]X-U_0[/imath] is not compact,or I don’t know the answer
But what does it mean (by definition) for [imath]U_0[/imath] to be open in [imath]X\cup \{p\}[/imath] ?