Property Suggestion: "manifold with boundary" and related properties

[prabau]

Property Suggestion

We currently have four properties related to being a manifold:

• (P124) topological n-manifold
• (P123) locally n-Euclidean
• (P122) locally Euclidean
• (P155) locally 1-Euclidean

For each of these, each point has a nbhd homeomorphic to some $\mathbb R^n$.

Another class of spaces are the "manifold with boundary", where each point has a nbhd homeomorphic to some closed upper half-space $\mathbb H^n=\{(x_1,\dots,x_n):x_n\ge 0\}\subseteq\mathbb R^n$.

So, at least for the first three properties above, we have corresponding

• (1) topological n-manifold with boundary
• (2) locally n-Euclidean-with-boundary
• (3) locally Euclidean-with-boundary

I just made up the names for the (2) and (3). Need to find good names.

(maybe "locally 1-Euclidean-with-boundary" also?)

Rationale

Manifolds with boundary are very common and not covered by usual manifolds.
Would allow to generalize various theorems we already have for locally Euclidean and the other properties.

Relationship to other properties

In particular:

• locally Euclidean-with-boundary => locally contractible.
  This would allow to derive for example that the intervals $[0,1]$ or $[0,1)$ or the telophase topology are locally contractible.

[prabau]

Need to find good names for (2) and (3).
I looked in Lee's Introduction to topological manifolds, and it does not define a "with-boundary" term corresponding to "locally n-Euclidean".

[prabau]

Note the special case for $n=0$. The space $\mathbb R^0=\{0\}$ is considered to be a singleton, and the corresponding $\mathbb H^0$ is set to be the same as $\mathbb R^0$ (Lee p. 43). So for the $0$-dimensional case, $0$-manifold with boundary is the same as $0$-manifold.

[Moniker1998]

I'm not convinced that we need locally euclidean with boundary.

[GeoffreySangston]

I think one of the main themes is likely to be to use the fact that a locally Euclidean space with boundary $M$ embeds into its double $2 M$, which is a locally Euclidean space. I.e., the double of Telophase topology S65 is Circle with two origins S209. I guess this would be a meta-property; analogs hold locally $n$-Euclidean with boundary and topological manifold with boundary (and locally $1$-Euclidean with boundary).

I'm visiting family today and tomorrow but I'll think more about this Issue thread soon.

[prabau]

Locally Euclidean-with-boundary applies to spaces that are not manifolds, because they are not Hausdorff or not second countable. For example, the long ray (S38). And for those, it's a convenient way to deduce they are locally contractible for example.

[Moniker1998]

I dislike the name. Maybe something like "locally a half-space" or similar would be better, if we're going to really implement this.

[prabau]

I am not attached to the name "locally Euclidean-with-boundary". From Google scholar, there are some expository notes about paracompact spaces that use it when describing the long ray. But not much else.

So by all means, we can brainstorm for a better name.

[prabau]

I dislike the name. Maybe something like "locally a half-space" or similar would be better, if we're going to really implement this.

Yeah, that's better. I would even prefer "locally a Euclidean half-space".
How do people feel about it (or something else)?

That's basically all I need before starting to work on the PR.

[Moniker1998]

@prabau I think that name is fine, and part of why is that it doesn't specify "closed". For each point there is a neighbourhood either homeomorphic to the closed or open half-space. Good that you have specified Euclidean - otherwise someone might question if we're considering half-spaces of Banach spaces, for example.

[prabau]

The main reason I was going with "locally a Euclidean half-space" without "closed" is not to make it too long already. People can look up the details in the property page when needed.

But, if a point has a nbhd homeomorphic to $\mathbb R^n$ (or an open set of $\mathbb R^n$), it also has a nbhd homeomorphic to the closed uppper half-space of $\mathbb R^n$ (where the point need not be on the "closed edge" of the nbhd). The full description could be "locally homeomorphic to a closed Euclidean upper half-place" (with "locally P" meaning each point has a (not necessarily open) nbhd with property P).

In the definition, we can expand this and mention that points fall into two categories: those with nbhd homeomorphic to some $\mathbb R^n$ and those not, blah blah closed edge blah blah...

[prabau]

So the names for the our properties would be:

• (1) topological $n$-manifold with boundary
• (2) locally an $n$-Euclidean half-space
• (3) locally a Euclidean half-space