Baire property is open subspace-closed
From Topospaces
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)
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Contents
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
Proof
Given: A Baire space , an open subset . A countable family of open dense subsets, of
To prove: The intersection is dense in
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Consider the set , where is the closure of in . is an open subset of . | the complement of a closed subset is open by definition. | |||
2 | Each is open in . | Fact (1) | is open in Each is open in . |
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3 | Each is open in with defined in Step (1). | Steps (1), (2) | Step-combination, and the observation that a union of open subsets is open | ||
4 | Each is dense in . In other words, for any open subset of , the intersection is nonempty. | [SHOW MORE] | |||
5 | Consider the collection of subsets . This is a countable collection of open dense subsets of | Steps (3), (4) | Step-combination direct | ||
6 | The intersection is dense in . | is a Baire space. | Step (5) | Step-given direct | |
7 | where | pure set theory | |||
8 | is dense in . | Steps (6), (7) | Step-combination direct | ||
9 | is dense in . In other words, for any nonempty open subset of , is nonempty | Fact (1) | is open in | 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