I know what a simplicial complex is, but when reading about triangulations on surfaces I found that there must exist a homeomorphism betwen the space underlying the surface and some simplicial complex. So my question is, how is defined the topology of a simplicial complex?
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What definition of simplicial complex do you know? – Cheerful Parsnip Nov 29 '13 at 22:03
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There is no (to my knowledge) topology directly on the simplicial complex. However, there is a way to canonically get a topological space from a simplicial complex called geometric realization. This is what you are looking for.
Najib Idrissi
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This works when the complex is finite. But an infinite complex in Euclidean space may not have the subspace topology. – Tim kinsella Nov 29 '13 at 20:55
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1@Timkinsella: What does not work, exactly? The construction I linked is perfectly valid and yields a topological space even when the complex is infinite. – Najib Idrissi Nov 29 '13 at 20:58
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Yes, you're right. Sorry about that. I still think the "coherent with its simplices" definition is simpler and more standard, though. – Tim kinsella Nov 29 '13 at 21:02
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@Timkinsella: It's exactly the same thing. Look closely, there is a direct limit involved in the definition of geometric realization. – Najib Idrissi Nov 29 '13 at 21:11
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Oh I agree they are equivalent. Sorry I didn't mean to nitpick. Two perspectives are always better than one :) – Tim kinsella Nov 29 '13 at 21:14
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Simplicial complexes have the topology coherent with their simplices (which are topologized as homeomorphs of the standard simplices living in Euclidean space). This means a subset of the complex is closed if and only if its intersection with each simplex is closed.
Tim kinsella
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