Proof that retract of Hausdorff space is closed

In summary, we can conclude that the final neighborhood in the proof is indeed disjoint from A, as explained by Harald Hanche-Olsen's comment.
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I think it is explained in Harald Hanche-Olsen's comment in reply to his own answer. It is the second comment to that answer, and dated Mar 6 '13 at 7:07. Except, I think he should have written ##y## instead of ##x## in it because he is referring to an arbitrary element of U, and ##x## has been fixed.To flesh it out a bit.
  1. We chose U and V to be disjoint nbds of x and a, which the Hausdorff property allows us to do.
  2. We know ##S\equiv r^{-1}(V\cap A)\cap U## is open because it is the intersection of two open sets. The first set is open because it is the pre-image of a set that is open in A under a continuous map r.
  3. Let ##y## be an arbitrary element of S. Then ##r(y)\in V## and ##y\in U## so, since ##U,V## are disjoint, we have ##r(y)\neq y##.
  4. Hence ##y\notin A## since every element of A maps to itself under ##r##.
  5. Hence S is an open nbd of ##x## that is disjoint from A.
  6. Since ##x## was chosen as an arbitrary point in ##X-A##, that means that every element of ##X-A## has an open nbd in ##X-A##.
  7. Hence ##X-A## is open. Hence ##A## is closed.
 
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We can show that the final neighborhood is disjoint from A by contradiction. Suppose the final neighborhood is not disjoint from A, then there exists a point x in both the final neighborhood and A. Since the final neighborhood is open, there exists an open set U containing x that is contained in the final neighborhood. But since x is also in A, U must also intersect A. This contradicts the fact that U is contained in the final neighborhood, which is supposed to be disjoint from A. Therefore, our initial assumption was incorrect and the final neighborhood is indeed disjoint from A.
 

1. What is a retract of a Hausdorff space?

A retract of a Hausdorff space is a subset of the space that can be continuously deformed onto a smaller subspace while preserving the original space's topology. In other words, it is a subspace that can be retracted onto a point without changing the topology of the original space.

2. Why is it important to prove that a retract of a Hausdorff space is closed?

Proving that a retract of a Hausdorff space is closed is important because it ensures that the topology of the original space is preserved when the subspace is retracted. This is a fundamental property of topological spaces that helps us understand the structure and behavior of these spaces.

3. How is the proof of a retract of a Hausdorff space being closed different from other topological proofs?

The proof of a retract of a Hausdorff space being closed is different from other topological proofs because it involves showing that the inverse image of any closed set in the subspace is also closed in the original space. This is a more specific and technical approach compared to other topological proofs.

4. Can you give an example of a retract of a Hausdorff space?

One example of a retract of a Hausdorff space is a circle within a sphere. The circle can be continuously deformed onto a single point, the center of the sphere, without changing the topology of the sphere.

5. Are there any exceptions to the rule that a retract of a Hausdorff space is closed?

No, there are no exceptions to this rule. It has been proven that any retract of a Hausdorff space is always closed. This is a fundamental property of topological spaces and holds true for all cases.

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