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, , , regularity, , complete regularity, and Tychonoffness are hereditary
Statement
Each of , , Hausdorffness (), regularity, , complete regularity, and Tychonoffness is hereditary.
Facts & Assumptions
Given: A subspace of a space with one of the listed separation properties.
, , and Hausdorffness are hereditary (, , and Hausdorffness are hereditary).
Regularity and complete regularity are hereditary (Regularity is hereditary, without a hidden hypothesis, Complete regularity is hereditary, without a hidden hypothesis).
means regular plus , and Tychonoff means completely regular plus (Regular spaces and spaces, with the source disagreement over whether regularity includes stated explicitly, Completely regular spaces and Tychonoff () spaces).
Proof
The first three assertions are [L1], and regularity and complete regularity are [L2].
A subspace of a space is regular by [L2] and by [L1], hence is .
A subspace of a Tychonoff space is completely regular by [L2] and by [L1], hence is Tychonoff.
These cover every property named in the statement.
Depends on
- $T_0$, $T_1$, and Hausdorffness are hereditary
- Regularity is hereditary, without a hidden $T_1$ hypothesis
- Complete regularity is hereditary, without a hidden $T_1$ hypothesis
- Regular spaces and $T_3$ spaces, with the source disagreement over whether regularity includes $T_1$ stated explicitly
- Completely regular spaces and Tychonoff ($T_{3\frac{1}{2}}$) spaces
Used by
- A worked preservation table for subspaces and products of the standard, cofinite, and ordinal topologies Example
- A non-hereditarily-Lindelof regular space yields an ideal witness Lemma
- Assuming countable choice, the deleted Tychonoff plank is a regular nonnormal open subspace of a compact Hausdorff normal space Lemma
- Preservation ledger for the separation axioms, with T₁ conventions kept explicit Remark
Dependency tree · two levels
20 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, §6 (standard reference, not scraped)
- Separation axiom (Wikipedia) (standard reference, not scraped)