What is the topology of a simplicial complex?

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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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2 Answers

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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.

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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.

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