# About simplicial complex

Let $K$ be a simplicial complex and if $\sigma$ is a simplex of $K$.

How to prove that the following sets are simplicial complexes?

1) The boundary of $K$: $\partial(K)=\{\mbox{proper faces which belong to all the simplexes of K}\}$

2) The closure of $\sigma$: $\mathrm{Cl}(\sigma)=\{\mbox{faces of }\sigma\}$

Can you help me please? Thank you!!!

-

2
What definition do you have of simplicial complexes? – Dedalus Nov 24 '12 at 19:36
I have to prove this definition: A simplicial complex K is a finite collection of simplices in some R^n satisfying: 1. If σ ∈ K, then all faces of σ belong to K. 2. If σ, τ ∈ K, then either σ ∩ τ = ∅ or σ ∩ τ is a common face of σ and τ. – lauren Nov 25 '12 at 8:27
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8344354629516602, "perplexity_flag": "middle"}