Hi, I would appreciate some guiding with this question. Thanks.

Let X be some topological space and Y a compact, Hausdorff topological space. Let and define .

Show that if is a closed subset of with the product topology, then f is continuous.

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- March 1st 2010, 02:23 PMDefunktTopology question
Hi, I would appreciate some guiding with this question. Thanks.

Let X be some topological space and Y a compact, Hausdorff topological space. Let and define .

Show that if is a closed subset of with the product topology, then f is continuous. - March 1st 2010, 02:36 PMDrexel28
- March 1st 2010, 03:01 PMDrexel28
**Theorem**: Let be any space and be compact. Then, if is closed we have that is continuous.

**Proof:**

**Lemma:**Let be compact, then is a closed.

**Proof:**Let be closed. We prove that is closed by proving that it's compliment is open. So, let . Then, it follows that , and for each has a basic open neighborhood . Since the tube is compact (since it's homeomorphic to ) there is a finite subcovering . The neighborhood is clearly disjoint from . The conclusion follows.

Now, it is easily proved that since is closed so is . But, it is clearly a bijective continuous mapping and since it's closed it's also a homeomorphism. The problem follows after considering the following lemma

**Lemma:**A mapping is continuous if is a homeomorphism.

**Proof:**We must merely note that . - March 1st 2010, 03:05 PMDefunkt
I think I came up with a solution. Tell me if you see anything wrong please.

I want to show that f is continuous, ie. that if is open then is open as well.

Let V be an open subset of Y. Let , then U is closed, and so is . Since is closed as well, is also closed.

From a previous result, we know that the projection is a closed mapping when Y is compact. Therefore, is closed.

However, note that is exactly the set of points in X which f does not map into V, and it is closed, therefore is open and we are done.

I know it is a bit informal, but that is simply the outline.

Thanks again for helping. - March 1st 2010, 03:16 PMDrexel28
- March 1st 2010, 03:29 PMDefunkt
- March 1st 2010, 03:31 PMDrexel28