Notes

Description: This note contains results from McCarten [1988], and Schnare [1968] relating to forms [1 BF] \((DT_0)\), [1 BG] \((CT_0)\), [1 BH] \((TT_0)\), and [1 BI] \((MT_0)\).

Content:

This note contains results from McCarten [1988], and Schnare [1968] relating to forms [1 BF] \((DT_0)\), [1 BG] \((CT_0)\), [1 BH] \((TT_0)\), and [1 BI] \((MT_0)\).

[1 BF] \((DT_0)\): Every topological space \(X\) has a \(T_0\) subspace that is dense in \(X\). (\(Y\) is dense in \(X\) if there is no non-empty open subset \(O\subseteq X\) such that \(O\cap Y=\emptyset\).)
[1 BG] \((CT_0)\): Every topological space \(X\) has a \(T_0\) subspace that is codense in \(X\). (\(Y\) is codense in \(X\) if there is no non-empty closed subset \(C\subseteq X\) such that \(C\cap Y=\emptyset\).)
[1 BH] \((TT_0)\): Every topological space \(X\) has a \(T_0\) subspace that is thick in \(X\). (\(Y\) is thick in \(X\) if there is no non-empty open and closed (clopen) subset \(H\subseteq X\) such that \(H\cap Y=\emptyset\).)
[1 BI] \((MT_0)\): Every topological space has a maximal \(T_0\) subspace.

Remarks:

  1. Forms \(DT_0\) and \(CT_0\) are false if ``\(T_0\)'' is replaced by ``\(T_1\)''. (A counter example is the topological space \((\Bbb R,\tau)\), where \(\tau=\{G\subseteq\Bbb R: G=(a,\infty), a\in\Bbb R \vee G=\emptyset\vee G=\Bbb R\}\).) McCarten [1988]
  2. [1 BH] \(TT_0\) remains equivalent to AC if \(T_0\) is replaced by \(T_n\), for any \(n\) such that \(T_n\) implies \(T_0\). McCarten [1988]
  3. \(MT_0\) is equivalent to \(MT_1\). Schnare [1968]
  4. Schnare [1968] shows that the statement \(MP\) is false if \(P = T_2, T_3, T_4\), regular, completely regular, normal, paracompact, metrizable, linearly disconnected or discrete. (It follows from the lemma that if \(P\) is a property that is possessed by every finite discrete space, but not by a countably infinite minimal \(T_1\) space, then \(MP\) is false.)

Howard-Rubin number: 106

Type: Summary of results

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