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I am a little confused about it. I need to prove that a simplicial complex is connected iff its 1-skeleton is path connected. Is it fine if I prove it for a CW complex since simplicial complex are special cases of CW complexes?

Thanks.

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    $\begingroup$ Yes it is fine. $\endgroup$
    – user87690
    Dec 7, 2013 at 17:19

1 Answer 1

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Yes, the geometric realization of a simplicial complex is a CW complex whose cells correspond to the simplices in the complex.

The connectedness of a CW complex only depends on the connectedness of the $1$-skeleton of the complex $X$. This is a special case of the general fact that the inclusion induces isomorphisms $\tilde H_k(X^n)\approx\tilde H_k(X)$ whenever $k<n$ (or if you are more familiar with homotopy groups, $\pi_k(X_n)\approx\pi_k(X)$). On the other hand, a direct proof is easy and is maybe more appropriate here.

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