# Monotonically normal implies hereditarily collectionwise 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 collectionwise normal space)

View all topological space property implications | View all topological space property non-implications

Get more facts about monotonically normal space|Get more facts about hereditarily collectionwise normal space

## Contents

## Statement

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

## Related facts

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

## Facts used

## Proof

**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.