Two definitions of locally compact

In summary, the proof shows that for a locally compact and Hausdorff space, there exists a simpler definition of locally compactness. This is proven by showing that for any compact set and point not in the set, there exists an open set containing the point and a compact set contained in the open set, such that the compact set and the original set are disjoint. The key step is showing that the compact set is contained in the open set.
  • #1
Fredrik
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I'm trying to understand the proof of (ii)[itex]\Rightarrow[/itex](i) of proposition A.6.2.(1) here. The theorem says that the given definition of "locally compact" is equivalent to a simpler one when the space is Hausdorff. I found the proof quite hard to follow. After a few hours of frustration I'm down to one last detail. Why is [itex]F\subset U_1[/itex]? It seems to me that F could contain limit points of [itex]U_1[/itex] that aren't in [itex]U_1[/itex].

Edit: I figured it out. The set [itex]V_2[/itex] is open in the topology of [itex]K_x[/itex], so there's an open set [itex]V_2'[/itex] such that [itex]V_2=K_x\cap V_2'[/itex]. This implies that

[tex]F=K_x-V_2=K_x-(K_x\cap V_2')=K_x-V_2'.[/tex]

We also have [itex]K_x-U_1\subset V_2\subset V_2'[/itex], and this implies that

[tex]K_x-V_2'\subset K_x-(K_x-U_1)=K_x\cap U_1\subset U_1.[/tex]
 
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  • #3


Hi there,

I see that you have been struggling with the proof for a while. I'm glad to hear that you finally figured it out!

Just to clarify for anyone else who might be reading, the proof is showing that if a space is locally compact and Hausdorff, then it satisfies a simpler definition of locally compactness. The key step is to show that for any compact set K and a point x not in K, there exists an open set U containing x and a compact set F contained in U such that K is disjoint from F.

The reason why F is contained in U is because of the definition of compactness. A compact set is one where every open cover has a finite subcover. In this case, the open cover for F is V_2, which is a subset of U. So by the definition of compactness, there exists a finite subcover of V_2, which is F. Therefore, F is contained in U.

I hope this helps clarify any confusion for others who might be reading. If you have any other questions, feel free to ask and I'll do my best to help.
 

1. What is the definition of locally compact?

The definition of locally compact refers to a mathematical notion in topology that describes a topological space as having certain properties at each point. Specifically, a locally compact space is one in which each point has a compact neighborhood.

2. What is the difference between the two definitions of locally compact?

The two definitions of locally compact are the open neighborhood definition and the one-point compactification definition. The open neighborhood definition states that a space is locally compact if every point has a compact neighborhood. The one-point compactification definition states that a space is locally compact if it is homeomorphic to the one-point compactification of a locally compact space.

3. Why are there two definitions of locally compact?

The two definitions of locally compact arise from the fact that different properties of a space can be used to define its local compactness. The open neighborhood definition is more intuitive and easier to apply, while the one-point compactification definition is more general and has broader applicability.

4. What are some examples of locally compact spaces?

Examples of locally compact spaces include Euclidean spaces, topological manifolds, and metric spaces. These spaces have the property that each point has a compact neighborhood, satisfying the open neighborhood definition of local compactness.

5. What are the applications of locally compact spaces?

Locally compact spaces have applications in various areas of mathematics, including functional analysis, differential geometry, and algebraic topology. They also have applications in physics, particularly in the study of quantum mechanics and quantum field theory.

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