# Definition:Zero Dimensional Space

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## Definition

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

Then $T$ is zero dimensional if and only if it has a basis whose sets are all both closed and open.

## Also known as

Some sources hyphenate: **zero-dimensional**.

## Also see

- Results about
**zero dimensional spaces**can be found here.

## Linguistic Note

The thinking behind applying the term **zero dimensional** to the concept of a **zero dimensional space** arises from the idea of the small inductive dimension in the context of the topological space.

The small inductive dimension of a topological space is defined to be $1$ greater than the small inductive dimension of its boundary.

By Set is Clopen iff Boundary is Empty, a topological space with a basis whose sets are all clopen has no boundary.

Thus it follows that such a topological space is considered to be **zero dimensional**.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Disconnectedness - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**zero-dimensional**(of a space)