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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
- Assuming countable choice, perfect normality, and hence T₆, is hereditary Corollary
- Complete normality, and hence T₅, is hereditary Corollary
- A space is Tychonoff if and only if it embeds in a cube [0,1]^J Theorem
- T₀, T₁, T₂, regularity, T₃, complete regularity, and Tychonoffness are hereditary Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 49 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)