Monotonically normal implies hereditarily collectionwise normal
This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., monotonically normal space) must also satisfy the second topological space property (i.e., hereditarily collectionwise normal space)
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- Monotonically normal implies collectionwise normal
- Monotonically normal implies hereditarily normal
- Monotonically normal implies normal
- Monotone normality is hereditary
Given: A monotonically normal space .
To prove: Every subspace of is a collectionwise normal space.
Proof: By fact (1), every subspace of is monotonically normal, and by fact (2), every subspace is thus collectionwise normal.