Proof of Topology: Compact Subsets in Open Sets

Thus, the statement holds in any topological space, not just regular spaces. In summary, the statement known as the generalized tube lemma holds in any topological space, not just regular spaces. This is because open sets in a product space are generated by sets of the form UxV, and any compact subset AxB can be contained in a finite union of sets of this form, which can then be contained in a single open set UxV.
  • #1
ForMyThunder
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If AxB is a compact subset of XxY contained in an open set W in XxY, then there exist open sets U in X and V in Y with AxB contained in UxV contained in W.

Is this true for all spaces XxY? Or does it hold for only regular spaces?
 
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  • #2
This statement is sometimes known as the generalized tube lemma. It holds in any topological space. Regularity is not required.
 
  • #3
This statement is true for all spaces XxY, not just regular spaces. The proof relies on the fact that open sets in a product space are generated by sets of the form UxV, where U is open in X and V is open in Y. Therefore, any compact subset AxB of XxY can be contained in a finite union of sets of this form, which can then be contained in a single open set UxV.
 

1. What is the definition of a compact subset in open sets?

A compact subset in open sets is a subset of a topological space that is both closed and bounded. In other words, it contains all of its limit points and can be contained within a finite number of open sets.

2. How is compactness related to the topology of a space?

Compactness is a topological property that describes the behavior of open sets in a space. It is related to the topology in that it characterizes the structure and behavior of open sets in a space, and can be used to distinguish between different topological spaces.

3. What is the importance of compactness in topology?

Compactness is an important concept in topology because it allows us to study the properties of a space without having to consider every single point individually. It also provides a way to compare and classify different spaces based on their topological properties.

4. How is compactness different from connectedness?

While compactness and connectedness are both topological properties, they are distinct concepts. Compactness refers to the behavior of open sets in a space, while connectedness refers to the relationship between points in a space. A space can be compact without being connected, and vice versa.

5. Can a subset of a compact space be non-compact?

Yes, it is possible for a subset of a compact space to be non-compact. This can happen if the subset is not closed or bounded, or if it contains limit points that are not in the original space. However, if the subset is closed and bounded, it will also be compact.

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