# (path)Connectedness of a simplicial complex Vs it's \$1\$-skeleton

by user   Last Updated September 12, 2019 04:20 AM

Let $$\Delta$$ be an abstract simplicial complex https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex and $$|\Delta|$$ be its geometric realization.

Let $$\Delta ^{(1)} :=\{F$$ is a face of $$\Delta : |F|\le 2 \}$$ . So $$\Delta ^{(1)}$$ is a simplicial complex of dimension $$1$$ , so in particular it is a simple graph. Now $$\Delta ^{(1)}$$ is connected as a graph iff its geometric realization $$|\Delta ^{(1)}|$$ is path connected. Now consider the following two statements:

(1) $$\Delta ^{(1)}$$ is connected as a graph.

(2) $$|\Delta |$$ is path connected as a topological space.

My question is: are (1) and (2) equivalent ? If not then at least, does one of them imply the other ?

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