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, , and Hausdorffness are hereditary
Statement
The properties , , and Hausdorffness are hereditary in the sense of Hereditary, open-hereditary and closed-hereditary properties of topological spaces.
Facts & Assumptions
Given: A subspace of a space carrying one of the stated properties.
distinguishes a distinct pair by one open set, separates each point from the other by an open set, and Hausdorffness separates a distinct pair by disjoint open sets ( (Kolmogorov) and (Frechet) spaces, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not).
Proof
Let be distinct. If is , choose an open containing exactly one of ; then does the same in .
If is , apply the preceding trace argument separately to the two open sets supplied by the condition, so each of has an open neighbourhood in missing the other.
If is Hausdorff, choose disjoint open containing respectively; and are disjoint open neighbourhoods in .
Since was arbitrary, each of the three properties is hereditary.
Depends on
- $T_0$ (Kolmogorov) and $T_1$ (Frechet) spaces
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- Hereditary, open-hereditary and closed-hereditary properties of topological spaces
Used by
- An injective immersion from a compact manifold is an embedding Corollary
- Assuming countable choice, perfect normality, and hence T₆, is hereditary Corollary
- Complete normality, and hence T₅, is hereditary Corollary
- Pullback connection Definition
- Vector field and section along a smooth curve Definition
- Endpoint-duplicating functions on [0,1] become all continuous functions on the endpoint quotient Example
- The polynomial algebra is dense but not closed on a nondegenerate compact interval Example
- An open subset of a smooth manifold has a canonical restricted smooth structure Proposition
- A space is Tychonoff if and only if it embeds in a cube [0,1]^J Theorem
- Compact Hausdorff Baire implies DMC Theorem
- T₀, T₁, T₂, regularity, T₃, complete regularity, and Tychonoffness are hereditary Theorem
Dependency tree · two levels
14 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
- J. P. May, An Outline Summary of Basic Point Set Topology, §§5–6 (standard reference, not scraped)
- Separation axiom (Wikipedia) (standard reference, not scraped)