Monotonically normal implies hereditarily collectionwise normal

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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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Any monotonically normal space is a hereditarily collectionwise normal space: every subspace of it is a collectionwise normal space.

Related facts

Facts used

  1. Monotone normality is hereditary
  2. Monotonically normal implies collectionwise normal


Given: A monotonically normal space X.

To prove: Every subspace of X is a collectionwise normal space.

Proof: By fact (1), every subspace of X is monotonically normal, and by fact (2), every subspace is thus collectionwise normal.