Topology/Perfect map

This is supposed to be a challenging section and will test the reader's understanding of the previous material.

Perfect maps are specific maps that have useful applications in point-set topology. A perfect map is a map which preserves 'inverse-like' properties. Like the continuous image of a connected space is always connected, if the perfect image (image under a perfect map) of a certain space, X, is connected, then X must be connected. Perfect maps are good in this sense as they are weaker than a homeomorphism, but strong enough to behave like homeomorphisms. We will give the formal definition in the next section.

Formal definition
Let X and Y be topological spaces and let p be a map from X to Y that is continuous, closed, surjective and such that p^(-1) {y} is compact relative to X for each y in Y. Then p is known as a perfect map.

Examples and Properties
The reader may wish to note that f is an open map if and only if the inverse of f is continuous.

1. If p: X->Y is a perfect map and if Y is compact, then X is compact.

2. If p: X->Y is a perfect map and if X is regular, then Y is regular (note that: if p is merely continuous, then if X is regular, Y need not be regular. An example of this is if X is a regular space and Y is an infinite set in the indiscrete topology)

3. If p: X->Y is a perfect map and if X is locally compact, then Y is locally compact.

4. If p: X->Y is a perfect map and if X is second countable, then Y is second countable.

5. Every injective perfect map is a homeomorphism. This follows from the fact that a bijective closed map has a continuous inverse.

6. If p: X->Y is a perfect map and if Y is connected, then X need not be connected. An example is when we let X be a compact disconnected space and Y be a single point topological space and p be the constant map.

7. A perfect map need not be open as the following map shows:

p(x) = x    if x belongs to [1,2]

p(x) = x-1  if x belongs to [3,4]

This map is closed, continuous (by the pasting lemma) and surjective and therefore is a perfect map (the other condition is trivially satisfied). However, p is not open for the image of [1,2] under p is [1,2] which is not open relative to [1,3] (the range of p). Note that this map is a quotient map and the quotient operation is ‘gluing’ two intervals together.

8. Notice how, to preserve properties such as local connectedness, second countability, local compactness etc… we require that the map be not only continuous but also open. A perfect map need not be open (see previous example), but these properties are still preserved under perfect maps.

9. Every homeomorphism is a perfect map. This follows from the fact that a bijective open map is closed and that since a homeomorphism is injective, the inverse of each element of the range must be finite in the domain (in fact, the inverse must have precisely one element).

Exercises
1. a) Prove that compactness is preserved under homeomorphisms; that is, if Y is homeomorphic to X and X is compact, then so is Y b) Prove property 1

2. a) Prove that regularity is preserved under homeomorphisms b) Prove property 2 (Hint: First prove the weaker result that if Y is Hausdorff so is X)

3. a) Prove that local compactness is preserved under homeomorphisms b) Prove property 3

4. a) Prove that second countability is preserved under homeomorphisms

b) Prove property 4

5. Determine whether or not the following theorem is valid. If you think that the theorem is valid, prove it. If not, find a counterexample:

Let X and Y be topological spaces and let p be a map from X to Y that is continuous, closed, surjective and such that p^(-1) {y} is CONNECTED relative to X for each y in Y. If Y is connected, then X is connected.

6. After studying the next section on quotient spaces, prove the following theorem:

Let q be a quotient map from X to Y such that q^(-1) {y} is connected for each y in Y. If Y is connected, then X is connected.