Is this a non-example of a $\Delta$-complex?

Question

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I know this is not a simplicial complex, but is it a $\Delta$-complex? Here is the definition of $\Delta$-complex from Hatcher:

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I don't believe any of the rules are broken.

Answer 1

This is not a $\Delta$-complex (at least not with the simplices drawn in the picture, with 2 triangles, 7 edges, and 8 vertices). It violates the second part of condition (i), that every point of $X$ is in the image of exactly one restriction $\sigma_{\alpha}|\mathring\Delta^n$. For instance, where the two triangles overlap, you have points that are in the image of the interiors of two different 2-simplices. Or, consider the dot in the middle of the right edge of the right triangle. That's a vertex, but it's also in the middle of an edge, so it is in the image of the interior of a 1-simplex and also in the image of the interior of a 0-simplex.

Answer 2

Certainly it can be given the structure of a delta complex, but the one suggested by the picture is not because it violates the first rule.