# Monotonically normal implies hereditarily normal

From Topospaces

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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## Contents

## Statement

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

## Related facts

- Monotone normality is hereditary
- Monotonically normal implies collectionwise normal
- Monotonically normal implies hereditarily collectionwise normal

## Facts used

## Proof

**Given**: A monotonically normal space .

**To prove**: Every subspace of is normal.

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