DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Totally disconnected spaces and totally separated spaces

Definition

Let $X$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).

The space $X$ is totally disconnected when every connected component of $X$ is a singleton; equivalently, each connected component in the sense of Connected components, quasicomponents, and totally disconnected spaces consists of just one point.

The space $X$ is totally separated when for every distinct $x, y \in X$ there is a clopen set $U \subseteq X$ with $x \in U$ and $y \notin U$ , where clopen means both open and closed and closedness is taken in the sense of Interior, closure, boundary, exterior, derived set and isolated point in a topological space.

Depends on

Used by

For compact Hausdorff spaces, total disconnectedness and total separatedness are equivalent Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)