Definition:Zero Dimensional Space

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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.