# Monotonically normal implies 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., normal space)

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## Statement

Any monotonically normal space is a normal space.

## Definitions used

### Monotonically normal space

`Further information: monotonically normal space`

A topological space is termed monotonically normal if it is a space and there exists an operator (called a **monotone normality operator**) from pairs of disjoint closed subsets to open subsets such that:

- For all disjoint closed subsets , is an open subset containing and whose closure is disjoint from .
- For closed subsets , if , , with disjoint and disjoint, we have .

### Normal space

`Further information: normal space`

A topological space is termed normal if it is a space and, for any two disjoint closed subsets and , there exist disjoint open subsets and such that contains and contains .

## Proof

**Given**: A monotonically normal space with a monotone normality operator .

**To prove**: is a normal space.

**Proof**: The -space part follows by definition. For the other part, note that and are disjoint open subsets containing and respectively.