Implications

We have the following indirect implication of form equivalence classes:

149 \(\Rightarrow\) 155
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
149 \(\Rightarrow\) 67 The axiom of choice in topology, Brunner, N. 1983d, Notre Dame J. Formal Logic
note-26
67 \(\Rightarrow\) 106 Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
106 \(\Rightarrow\) 78 Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
78 \(\Rightarrow\) 155 Geordnete Lauchli Kontinuen, Brunner, N. 1983a, Fund. Math.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
149:

\(A(F)\):  Every \(T_2\) topological space is a continuous, finite to one image of an \(A1\) space.

67:

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

106:

Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.

78:

Urysohn's Lemma:  If \(A\) and \(B\) are disjoint closed sets in a normal space \(S\), then there is a continuous \(f:S\rightarrow [0,1]\) which is 1 everywhere in \(A\) and 0 everywhere in \(B\). Urysohn [1925], pp 290-292.

155:  \(LC\): There are no non-trivial Läuchli continua. (A Läuchli continuum is a strongly connected continuum. Continuum \(\equiv\) compact, connected, Hausdorff space; and strongly connected \(\equiv\) every continuous real valued function is constant.)

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