# Closure of Intersection is Subset of Intersection of Closures

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## Contents

## Theorem

Let $T$ be a topological space.

Let $I$ be an indexing set.

Let $\forall i \in I: H_i \subseteq T$.

Then:

- $\displaystyle \left({\bigcap_I H_i}\right)^- \subseteq \bigcap_I H_i^-$

where $H_i^-$ denotes the closure of $H_i$.

## Proof

Since $\displaystyle \bigcap_I H_i^-$ is an intersection of closed sets, it is closed, from Topology Defined by Closed Sets.

Also, it contains $\displaystyle \bigcap_I H_i$ and so by the main definition of closure also contains $\displaystyle \left({\bigcap_I H_i}\right)^-$.

$\blacksquare$

## Also see

- Closure of Intersection may not equal Intersection of Closures, which shows that equality does not generally hold.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.7$: Definitions: Proposition $3.7.17$