Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
375 \(\Rightarrow\) 78 clear
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
375:

Tietze-Urysohn Extension Theorem: If \((X,T)\) is a normal topological space, \(A\) is closed in \(X\), and \(f: A\to [0,1]\) is continuous, then there exists a continuous function \(g: X\to [0,1]\) which extends \(f\).

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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