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): zero-dimensional (of a space)
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): zero-dimensional (of a space)