Questions about Covering maps, manifolds, compactness

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In summary, a covering map is a continuous function that "covers" one topological space with another, while a manifold is a topological space that is locally Euclidean. Compact spaces can be covered by a finite number of open sets, while non-compact spaces cannot. Covering maps and manifolds are often related, as covering maps can be used to study manifolds and many manifolds are themselves covering spaces. These concepts have various real-life applications in fields such as physics, engineering, computer graphics, and data analysis.
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PsychonautQQ
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Suppose ##p: C-->X## is a covering map.

a) If ##C## is an n-manifold and ##X## is Hausdorff, show that ##X## is an n-manifold.
b) If ##X## is an n-manifold, show that ##C##is an n-manifold
c) suppose that ##X## is a compact manifold. Show that ##C## is compact if and only if p is a finite sheeted covering.Thoughts:

a) I will have to show that ##X## is second countable and has a basis of n-euclidean balls. To show that it has a basis of n-euclidean balls, I could take the pre image of any basis element of ##X## and then use any component of the preimage which will be homeomorphic to both a euclidean n-ball and the basis element in ##X##. To show that ##X## is second countable I don't know what I would do yet.

b) The key here is using the fact that slices of the preimage of ##p## will map homeomorphically to their source neighborhood in ##X##. Perhaps if I can show that the preimage of basis elements of ##X## is surjective onto ##C## the proof will not be far away.

c) I could suppose ##C## is compact and then proceed by contradiction: Take the preimage of any neighborhood in ##X## and use this infinite preimage to construct an open covering of ##C## with no finite sub covering. I am hesitant about this strategy because I haven't seen how I can use the fact that ##X## is a compact manifold.

Any feedback is appreciated as always! Thanks PF!
 
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For (a) and (b) we need to prove the second-countability and locally Euclidean properties. For (b) we also need to prove C is Hausdorff. So, five things to prove overall between the two sub-questions. I found three of them not too hard.

For (a) a natural suggestion for a countable basis for X would be ##\mathscr B^X=\{f(B)\ :\ B\in\mathscr B^C\}## where ##\mathscr B^C## is a countable basis for C. Then take an open set U in X and show it can be written as a union of elements of ##\mathscr B^X##.

My proof of that uses the axiom of choice to first express X as a union of open sets each of which is homeomorphic to a stack of homeomorphic images in C. From there on it is easy. I am pondering whether there's a version that doesn't use Choice.

For (b) the hard bit will be showing second-countability. The trouble with just considering pre-images of basis elements of X is that, if we split them up into their connected components, there may be uncountably many since, as I recall, there is nothing in the definition of covering space to say that the stack of slices above an open set in X can't be uncountable. But if we don't split them up then we may lose Hausdorffness of C since we may no longer be able to separate different slices by open sets.

So I currently have no answer for second-countability in (b) and only a Choice-tainted answer for second-countability in (a). But I can prove the other bits. Let me know if you want any hints for them.

I haven't thought about (c) yet. 'If and only if' questions make me feel tired, because one has to prove BOTH directions.
 
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a) To show that X is second countable, we can use the fact that C is an n-manifold and thus has a countable basis of n-euclidean balls. Then, since p is a covering map, for any x in X, there exists an open neighborhood U of x that is evenly covered by p. By the definition of a covering map, this means that the preimage of U under p is a disjoint union of open sets in C, each of which is homeomorphic to U. Since C has a countable basis of n-euclidean balls, the preimage of U can be written as a countable union of n-euclidean balls, which forms a basis for X. Thus, X is second countable.

b) To show that C is an n-manifold, we can use the fact that X is an n-manifold and p is a covering map. Since X is an n-manifold, for any x in X, there exists an open neighborhood U of x that is homeomorphic to an open subset of R^n. Since p is a covering map, the preimage of U under p is a disjoint union of open sets in C, each of which is homeomorphic to U. Thus, for any x in C, there exists an open neighborhood V of x that is homeomorphic to an open subset of R^n. Therefore, C is an n-manifold.

c) Suppose p is a finite sheeted covering. This means that for any x in X, the preimage of x under p has a finite number of points. Since X is compact, any open covering of X has a finite subcovering. Therefore, if we take an open covering of C and map it to X using p, the finite number of points in the preimage of each point in X will give us a finite subcovering of C. Thus, C is compact.

Now suppose p is not a finite sheeted covering. This means that there exists a point x in X whose preimage under p is infinite. Since X is compact, any open covering of X must have a finite subcovering. However, if we take an open covering of C and map it to X using p, the infinite preimage of x will prevent us from finding a finite subcovering of C. Therefore, C is not compact.
 

Related to Questions about Covering maps, manifolds, compactness

1. What is a covering map?

A covering map is a continuous function between topological spaces that has the property of "covering" one space with another. This means that for every point in the first space, there exists an open neighborhood that can be mapped homeomorphically onto an open neighborhood in the second space.

2. What is a manifold?

A manifold is a topological space that is locally Euclidean, meaning that for each point there exists a neighborhood that is homeomorphic to an open subset of Euclidean space. Manifolds come in different dimensions, such as 1-dimensional curves, 2-dimensional surfaces, and 3-dimensional volumes.

3. What is the difference between a compact space and a non-compact space?

A compact space is a topological space that is "small" in a sense that it can be covered by a finite number of open sets. In contrast, a non-compact space is "big" and cannot be covered by a finite number of open sets. Compact spaces have many useful properties, such as being closed and bounded, and are often easier to work with in mathematical proofs.

4. How are covering maps and manifolds related?

Covering maps are often used to study manifolds, as they provide a way to "cover" a manifold with simpler, more well-understood spaces. This allows for easier analysis and understanding of the manifold's properties. Additionally, many manifolds are themselves covering spaces of other spaces, making covering maps a crucial tool in the study of manifolds.

5. What are some real-life applications of covering maps and manifolds?

Covering maps and manifolds have numerous applications in fields such as physics, engineering, and computer graphics. For example, they are used in the study of fluid dynamics and electromagnetism, as well as in the design of 3D models for animation and video games. In addition, covering maps and manifolds are used in data analysis and visualization to understand complex data sets and relationships between variables.

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