What is meant by "locally like a simplical complex?"

[The Bill]

What is the name for a toplogical space that is everywhere "locally like a simplical complex" in that every point has at least one neighbourhood which is either a topological manifold, or can be countably decomposed by surgery into a set of topological manifolds which intersect along submanifolds?

Basically, what I'm looking for is a topological generalization for things like the structure of the surfaces in a foam of soap bubbles, but where the dimensions of the components aren't always the same across the space.

 

[The Bill]

The thread title is wrong, I didn't mean to mean what it says.

I can't edit it now, but I meant something like:

What term means "locally like a simplical complex?"

 

[fresh_42]

Have you considered CW-complexes whether they match your requirements?
https://en.wikipedia.org/wiki/CW_complex

 

[The Bill]

If every point in a topological space X has at least one neighborhood which is homeomorphic to a CW complex, does that necessarily imply that X is homeomorphic to a CW complex? If it does, then yes, CW complexes could be what I'm looking for.

Edit, of course, then I'd want a single word adjective that means "is a CW complex" to answer the thought I had which lead to this thread.

 

[fresh_42]

... in that every point has at least one neighbourhood which is either a topological manifold...

Isn't that true for every manifold, since you can always regard an open neighborhood as a manifold itself simply by restriction?
What you're looking for reads as a bundle of topological spaces. I thought a CW-complex as a generalization of simplicial complexes has at least a bit of a structure. My first thought as I read your post has been a cell-complex.

 

[The Bill]

Isn't that true for every manifold, since you can always regard an open neighborhood as a manifold itself simply by restriction?
What you're looking for reads as a bundle of topological spaces. I thought a CW-complex as a generalization of simplicial complexes has at least a bit of a structure. My first thought as I read your post has been a cell-complex.

I don't think "at least one neighborhood of every point in X has property Foo" implies "X has property Foo" no matter what property Foo is. I'm just saying that while what I'm asking might be obvious to some people, I can't think of a reason that it's trivial.

 

[fresh_42]

Nor have I said it's trivial, neither have you said implies. You said has. And I said implicitly that it is a too vague description. The comparison with the foam has been more precise than the topological definition. But you are right, perhaps someone knows right away a space that is at each point a simplicial complex without necessarily being one itself.

 

[micromass]

I'm not sure this is what you want, but it seems you're interested in a kind of "manifold" but where you allow singularities. This is where the concept of analytic space becomes important: https://en.wikipedia.org/wiki/Analytic_space

So any simplicial complex can be seen (locally at least) as the intersection and union of hyperplanes, planes, lines and points. Now it happens to be that those things are exactly solutions to polynomial equations. I think the converse is also true by Hironaka, that every solution to a polynomial equation is locally triangulable.

So perhaps the answer is the underlying topological space of an analytic space/variety?

 

[disregardthat]

You have the example of the double point on a line: two copies of the real number line R glued along R\0. This space is locally euclidean, hence locally a simplicial complex, but is not a hausdorff space. So it cannot be a CW-complex.

Note also that there exists manifolds which are not simplicial complexes. That would be a stronger counterexample

 

[micromass]

Note also that there exists manifolds which are not simplicial complexes. That would be a stronger counterexample

But those are of course all locally simplicial complexes. No?

 

[disregardthat]

But those are of course all locally simplicial complexes. No?

Yes, that was my point. It's not strictly stronger though, just another example of the fact that locally X does not imply globally X in this context. To sum up: locally euclidean spaces need not be CW-complexes, and compact manifolds (which actually are CW complexes) need not be simplicial complexes.

Just to clarify further: Locally euclidean implies locally a simplicial complex, and all manifolds are locally euclidean.

I also think that (real) analytic spaces may be exactly what The Bill is looking for.

 

[The Bill]

So, what I'm looking for is an adjective that provides a way to say "is homeomorphic to the underlying topological manifold of a real analytic space," in a more straightforward and compact way.

 

[micromass]

I don't know the answer but a real analytic space is not generally a topological manifold!

I've read terms like "locally triangulable". Maybe that is what you want?

 

[fresh_42]

So, what I'm looking for is an adjective that provides a way to say "is homeomorphic to the underlying topological manifold of a real analytic space," in a more straightforward and compact way.

May I ask why?

 

[The Bill]

May I ask why?

Economy of language.

 

[mathwonk]

at the beginning of your paper you could say, we will use the terminology LS for a space which is locally simplicial.. Then you have a short term for it.

 

[disregardthat]

Or you could say "In this paper we will assume all spaces to be locally simplicial". That way you don't even have to mention it.