# Definition:Dense-in-itself

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*This page is about Dense-in-itself in the context of topology. For other uses, see Dense.*

## Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

Then $H$ is dense-in-itself if and only if it contains no isolated points.

## Also see

- Results about
**topological denseness**can be found here.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Limit Points

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**dense set**