Open subspace of a compact space topological space

In summary, the fact that a compact topological space X has a closed subspace that is compact does not necessarily mean that an open subspace G of X is also compact. This can be seen through a counterexample, such as the open interval (0,1) contained in the closed interval [0,1]. Therefore, it is not true that every open subspace of a compact space is compact.
  • #1
de_brook
74
0
It is a fact that if X is a compact topoloical space then a closed subspace of X is compact.
Is an open subspace G of X also compact?
please consider the following and note if i am wrong;

proof: Since G is open then the relative topology on G is class {H_i}of open subset of X such that the union of all sets in this class is G. but X is compact and each H_i is the intersection of G with an open subset P_i of X for corresponding i. The result follows from the fact {p_i} has a finite subclass which contains X.
hence every open subspace of a compact space is compact.

pls, am i right?
 
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  • #2
Your proof is wrong, as you can have an open cover of G in the topology of X that doesn't cover X, and then when you reduce it to the induced topology on G it doesn't need to have a finite subcover.

Conclusion: All you did is prove that an open cover of a compact space is also an open cover of its subsets (not a very impressive result :p )

As a quick counterexample, see (0,1) contained in [0,1]
 

1. What is an open subspace of a compact space topological space?

An open subspace of a compact space topological space is a subset of a topological space that is open in the sense of the topology induced by the compact space. This means that for every point in the subspace, there is an open set containing that point that is also contained within the subspace.

2. How is an open subspace different from a closed subspace?

An open subspace is a subset that is open in the topology of the larger space, while a closed subspace is a subset that is closed in the topology of the larger space. This means that the complement of an open subspace is closed, and the complement of a closed subspace is open.

3. What is the significance of an open subspace in a compact space topological space?

An open subspace is significant because it allows us to study the properties of a compact space in a smaller, more manageable subset. This can make it easier to prove certain theorems or analyze certain aspects of the space.

4. Can an open subspace of a compact space topological space also be compact?

Yes, an open subspace of a compact space topological space can also be compact. This is because a compact subspace of a compact space is also compact.

5. How is the open subspace topology related to the original topology of the compact space?

The open subspace topology is a restricted version of the original topology of the compact space. It includes only those sets that are open in the larger space and also contained within the subspace. In other words, it is the topology induced by the subspace.

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