0

Let $f:N \to M$ be a continuous function between general topological spaces. Let's assume that each point $p \in N$ has an open neighborhood $U$ such that $f|_U$ is an injection. Then $f|_U$ is a continuous bijection onto its image. However, $f|_U$ may not be a homeomorphism, so we cannot expect $f$ to be a topological immersion in general. If $N$ and $M$ are topological manifolds (Hausdorff, second countable, locally Euclidean), is $f$ a topological immersion? I know that, if $M$ is Hausdorff and we can take compact $V \subset U$ for each $p$ then $f|_V$ must be a homeomorphism onto its image, and so $f$ is a topological immersion. I conjectured that it should be for manifolds but failed to argue.

cleone
  • 449
  • What is your definition of a topological immersion between topological spaces? – Moishe Kohan Feb 14 '17 at 10:46
  • The definition is a continuous map for which each point of the domain has a neighborhood that is mapped onto its image homeomorphically. – cleone Feb 14 '17 at 11:24
  • Oh, I see. Then all what you need is that $N$ is locally compact and $M$ is Hausdorff. Now, note that manifolds are locally compact. Incidentally, there is no consensus on what a "neighborhood" means. Some people require it to be open, some do not. – Moishe Kohan Feb 14 '17 at 11:29
  • I'm sorry. My question had a bug. My neighborhood is an open set. I edited the question to be clear. In locally compact spaces, each point has a neighborhood $V$ whose closure is compact. But how can I sure that $\bar{V}$ is contained in the open set $U$ in my question? – cleone Feb 14 '17 at 11:47
  • The most common definition of "locally compact" is that every point has a basis consisting of compact neighborhoods: Even open neighborhood $U$ of $x$ contains a compact neighborhood of $x$. This is what you want to use. – Moishe Kohan Feb 14 '17 at 11:50
  • @MoisheCohen Thank you. Now I understand it. – cleone Feb 14 '17 at 12:03
  • 1
    Isn't the definition proposed in the second comment precisely that of a local homeomorphism? – Arrow Dec 28 '17 at 19:24
  • @MoisheKohan Any response to Arrow please? –  Jun 27 '19 at 10:34
  • 1
    @SeleneAuckland Arrow is wrong, if you want a response. – Moishe Kohan Jun 27 '19 at 11:17
  • @Arrow And what about you please? –  Jun 28 '19 at 08:09
  • @MoisheKohan Okay thanks. I may think about this more later. –  Jun 28 '19 at 08:09
  • 1
    @SeleneAuckland Just take a map from a singleton to the real line. – Moishe Kohan Jun 28 '19 at 08:37
  • 1
    My comment was wrong. I asked a related question here. – Arrow Jun 28 '19 at 09:50

1 Answers1

1

First of all, there are different notions of topological immersions:

1) A continuous locally injectve map between topological spaces. (Regretfully, one can find this in Spivak.)

2) A continuous map $f: X\to Y$ such that for every $x\in X$ there exists a neighborhood $U$ of $x$ such that $f|_U: U\to f(U)\subset Y$ is a homeomorphism.

3) Assuming that $X, Y$ are topological manifolds: A continuous map $f: X\to Y$ such that for every $x\in X$ there exists a neighborhood $U$ of $x$ and a neighborhood $V$ of $y=f(x)$ and homeomorphisms $\phi: U\to R^m, \psi: V\to R^n$ such that the composition $$ \psi\circ f \circ \phi^{-1} $$ is an injective linear map $R^m\to R^n$. This you can find in the "Manifold Atlas" and it is my favorite definition. Unlike 1 and 2, it allows one to prove some nontrivial theorems. An example which satisfies 2 but not 3 is a wild sphere in $R^3$. Maybe one should call maps satisfying 3 "tame topological immersions".

As for a relation between 1 and 2, they become equivalent if $X$ is locally compact and $Y$ is Hausdorff. Unfortunately, again, there are two closely related but inequivalent notions of local compactness. My favorite definition is that every point admits a basis consisting of compact neighborhoods, where neighborhoods are understood in the sense of Bourbaki. Another common definition of local compactness is that every points admits a compact neighborhood (or a relatively compact neighborhood if one requires neighborhoods to be open). The two definitions are equivalent if the space is assumed to be Hausdorff. See this Wikipedia page for further discussion. The bottom line is that if $X$ and $Y$ are Hausdorff and $X$ is locally compact (in either sense) then 1 is equivalent to 2.

Lastly, there is the notion of a local homeomorphism, which is a map $f: X\to Y$ such that every $x\in X, y=f(x)\in Y$ admit neighborhoods $U, V$ such that $f|_U: U\to V$ is a homeomorphism. This is, of course, strictly stronger than 2 or 3, but is equivalent to 3 if $X, Y$ are manifolds of the same dimension.

Moishe Kohan
  • 97,719