My question is as the title suggests:
Which topological spaces can be realized as simplicial complexes (possibly abstract simplicial complex)?
A secondary question is which spaces can be realized as finite simplicial complexes?
I am aware of certain values which are defined for simplicial complexes, and wonder whether one can define them on topological space if they can be realized as a simplicial complex. My primary motivating example is the Euler characteristic, which is also defined on manifolds. To this end I was wondering which topological spaces could we try to extrapolate the definition of Euler characteristic?
I suspect that simplicial complexes can only be used to realize locally euclidean spaces, but perhaps some here happens to know the answer?