A subspace of a Hausdorff space is Hausdorff

A subspace of a Hausdorff space is Hausdorff.

Let X be Hausdorff and

Since Y is a subspace, where C is some open set in X.

Let

is the intersection of open sets so it is open. Therefore, where U and V are open sets in . Again since is open, U and V can be constructed in such a manner that .

Thus, the subspace is Hausdorff.

Correct?

Re: A subspace of a Hausdorff space is Hausdorff

Quote:

Originally Posted by

**dwsmith** A subspace of a Hausdorff space is Hausdorff.

Let X be Hausdorff and

Since Y is a subspace,

where C is some open set in X.

Let

is the intersection of open sets so it is open. Therefore,

where U and V are open sets in

. Again since

is open, U and V can be constructed in such a manner that

.

Thus, the subspace is Hausdorff.

Correct?

This isn't quite correct. The fact that is a subspace of says that given any open subset one has that for some open set [etx]V[/tex] in . want to take this and try again?

Re: A subspace of a Hausdorff space is Hausdorff

Quote:

Originally Posted by

**dwsmith** A subspace of a Hausdorff space is Hausdorff.

Let X be Hausdorff and

Since Y is a subspace,

where C is some open set in X.

Suppose that . Then .

But that means **there are two disjoint open sets** in such that WHY?

What can you say about

Re: A subspace of a Hausdorff space is Hausdorff

Quote:

Originally Posted by

**Drexel28** This isn't quite correct. The fact that

is a subspace of

says that given any open subset

one has that

for some open set [etx]V[/tex] in

. want to take this and try again?

From that, we know U is open in X since it is the intersection of two open sets in X. Let . Since U is open, disjoint sets can be constructed s.t. and

Does this do it then?

Re: A subspace of a Hausdorff space is Hausdorff

Re: A subspace of a Hausdorff space is Hausdorff

Re: A subspace of a Hausdorff space is Hausdorff

Quote:

Originally Posted by

**dwsmith** Let

;

, and

where

are open in X.

.

Yes.

Correct?

Yes.