Monotonically normal implies hereditarily 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 normal space)
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Any monotonically normal space is a hereditarily normal space: every subspace of the space is a normal space.

Related facts

Facts used

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


Given: A monotonically normal space X.

To prove: Every subspace of X is normal.

Proof: By fact (1), every subspace of X is monotonically normal. Using fact (2), we obtain that every subspace of X is normal.