Quote:

"We define an abstract simplicial complex to be a collection K of nonempty finite sets called (abstract) simplices, subject only to one condition: If s is in K, then every nonempty subset of s is in K."

I try to relate this to the concept of an euclidean simplicial complex, but it doesn't make any sense to me.

For example:

Consider the 1-simplex spanned by two vertices (0) and (1). The 1-simplex is then the closed interval [0,1].

Let K be the euclidean simplicial complex with sets {(0), (1) and [0,1]}, i.e. with three elements.

Om ok with this so far, but then K fails to be an abstract simplicial complex since the interval (1/3, 2/3) is not in the set.

(The definition clearly states that (1/3, 2/3) should be in the set as it is a nonempty subset of [0,1])

This seems odd as my intuition tells me that abstract simplicial complexes should be a generalization of euclidean simplicial complexes and this would thus be a counterexample.

I understand that some part of my reasoning must be wrong, but I cannot see in what way.

Can anyone please help me and try to explain?