Baire property is open subspace-closed

From Topospaces
Jump to: navigation, search
This article gives the statement, and possibly proof, of a topological space property (i.e., Baire space) satisfying a topological space metaproperty (i.e., open subspace-closed property of topological spaces)
View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions
Get more facts about Baire space |Get facts that use property satisfaction of Baire space | Get facts that use property satisfaction of Baire space|Get more facts about open subspace-closed property of topological spaces

Statement

Verbal statement

Every open subset of a Baire space is itself a Baire space, under the subspace topology.

Definitions used

Baire space

Further information: Baire space

A topological space is termed a Baire space if an intersection of countably many open dense subsets is dense.

Subspace topology

Further information: Subspace topology

Facts used

  1. Open subset of open subspace is open

Proof

Given: A Baire space X, an open subset A. A countable family of open dense subsets, U_n, n \in \mathbb{N} of A

To prove: The intersection T = \bigcap_{n \in \mathbb{N}} U_n is dense in A

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 Consider the set B = X \setminus \overline{A}, where \overline{A} is the closure of A in X. B is an open subset of X. the complement of a closed subset is open by definition.
2 Each U_n is open in X. Fact (1) A is open in X
Each U_n is open in A.
3 Each U_n \cup B is open in X with B defined in Step (1). Steps (1), (2) Step-combination, and the observation that a union of open subsets is open
4 Each U_n \cup B is dense in X. In other words, for any open subset V of X, the intersection (U_n \cup B) \cap V is nonempty. [SHOW MORE]
5 Consider the collection of subsets U_n \cup B. This is a countable collection of open dense subsets of X Steps (3), (4) Step-combination direct
6 The intersection \bigcap_{n \in \mathbb{N}} (U_n \cup B) is dense in X. X is a Baire space. Step (5) Step-given direct
7 \bigcap_{n \in \mathbb{N}} (U_n \cup B) = T \cup B where T = \bigcap_{n \in \mathbb{N}} U_n pure set theory
8 T \cup B is dense in X. Steps (6), (7) Step-combination direct
9 T is dense in A. In other words, for any nonempty open subset S of A, S \cap T is nonempty Fact (1) A is open in X Step (8) [SHOW MORE]

References

Textbook references

  • Topology (2nd edition) by James R. Munkres, More info, Page 297, Lemma 48.4, Chapter 4, Section 48