# Definition:Proper Subset/Improper

< Definition:Proper Subset(Redirected from Definition:Improper Subset)

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## Definition

Let $T$ be a set.

$S$ is an **improper subset** of $T$ if and only if $S$ is a subset of $T$ but specifically *not* a proper subset of $T$.

That is, either:

- $S = T$

or:

- $S = \O$

## Also defined as

Some sources categorise the empty set $\O$ as a proper subset, and not an **improper subset**.

As this is merely a matter of nomenclature, this distinction should not be of great importance.

However, it is wise to make sure which usage is intended when it is encountered.

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets