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(Cross-posted from MSE: https://math.stackexchange.com/questions/4425225/definition-of-union-of-simplicial-complex-and-a-subset)

Consider a simplicial complex $\Delta$ with vertex set equal to some finite set $E$. Once in a while, I came across some expressions for simplicial complexes of the form $\Delta \cup \alpha$ for some subset $\alpha \subset E$ usually not belonging to $\Delta$. How is this expression generally defined?

Some guesses I had were unions of the form $\gamma \cup \alpha$ for some subset $\gamma \subset E$ belonging to $\Delta$ or adding $\alpha$ to the collection of subsets belonging to the new simplicial complex. Is there a specific standard definition? The only definitions involving unions and simplicial complexes that I've seen written out in full are those of joins of simplicial complexes (which use unions of subsets belonging to the respective simplicial complexes).

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    $\begingroup$ It would be a lot easier to guess the meaning given context. $\endgroup$
    – Reid Barton
    Commented Apr 12, 2022 at 15:06
  • $\begingroup$ This came up in the statement I read that $I(G - e) = I(G) \cup \{ e \}$, where $I(G)$ is the independence complex (independent vertices) of a graph $G$ and $G - e$ means deleting the edge $e$ of $G$ keeping the vertices in $G$. It seemed a bit strange I'm not sure if the statement above is consistent with the definition of an independent set of vertices (e.g. not splitting into cases depending on the reason why some set of vertices is independent in $G - e$ and not in $G$) . Aside from this statement, I was wondering if there was a more general notation for the union in the question. $\endgroup$
    – modnar
    Commented Apr 12, 2022 at 18:20

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