Continuous bijection that is not an embedding

[Bacle]

Hi:
Just curious: a continuous function f:X-->Y ; X,Y topological spaces, can fail
to be an embedding because it is not 1-1, or, if f is 1-1 , f can fail to be an
embedding because, for U open in X f(U) is not open in f(X).

Can anyone think of a "reasonable" example of the last case, by reasonable
I mean no spaces with finitely many points, the inclusion map from a space
into itself with a different topology (i.e.: we have (X,T) and (X,T') , and we
use i:X-->X : i(x)=x ) , or maybe or something one could find in
"Counterexamples" book.

If A is a strict subspace of X , then the subspace topology on A guarantees
that i:A-->X , the inclusion, is an embedding. Is this true for other topologies on A.?

Thanks.

 

[g_edgar]

X = interval [0,2 pi), Y = circle {(x,y) : x^2+y^2=1}, continuous bijection f(t) = (cos(t),sin(t)).

 

[g_edgar]

>> If A is a strict subspace of X , then the subspace topology on A guarantees
that i:A-->X , the inclusion, is an embedding. Is this true for other topologies on A.?

No. In fact, that is the definition of "subspace topology" ... so for any topology on A other than the subspace topology, the inclusion i is NOT an embedding. Either because i is not continuous, or because i is not open, or both.

 

[Bacle]

Edgar wrote
" No. In fact, that is the definition of "subspace topology" ... so for any topology on A other than the subspace topology, the inclusion i is NOT an embedding. Either because i is not continuous, or because i is not open, or both. "

Thanks, Edgar. I thought the subspace topology was the initial topology with respect
to inclusion, and, given an inclusion:

i:A-->X

the initial topology was (is) , by def., the smallest topology on A, which makes the
inclusion continuous (largest topology being the discrete one, 2^A), but
I don't see how it follows that the subspace topology is the only one for which i is
( I am.? :) ) an embedding:


We have A< X a strict subspace ( i.e, A is not all of X ). Given the map : i:A-->X

We want to define a topology on A such that :

1) Continuity of i: For U open in X , i^-1(U) open in A.


We have that i^-1(U) =U/\A . So we must have U/\A open in A.

Then any topology T_A on A must contain the subspace topology.

i.e., T_A> (A, subspace) (with > meaning contains)


2) Openness of map i: For V open in A, i(V)=V is open in X . Like you said,
it follows that A must be an open subspace of X, i.e., A is open as a subset
of X.

How does it then follow that the only topology that makes i:A-->X
into an embedding is (A, subspace) .?. I don't see how we can conclude,
e.g.,
(A, subspace)> T_A

 

[Landau]

Instead of `No. In fact, that is the definition of "subspace topology"', I think g_edgar should have written `No. In fact, that is the definition of "topological embedding"'.

Let X,Y be topological spaces. A function f:X->Y is an embedding if
* f is injective, and
* f is a homeomorphism onto its image f(X), where f(X) carries the subspace topology of X.

So the requirement of "subspace topology" is part of the definition of embedding.

 

[RedX]

Instead of `No. In fact, that is the definition of "subspace topology"', I think g_edgar should have written `No. In fact, that is the definition of "topological embedding"'.

Let X,Y be topological spaces. A function f:X->Y is an embedding if
* f is injective, and
* f is a homeomorphism onto its image f(X), where f(X) carries the subspace topology of X.

So the requirement of "subspace topology" is part of the definition of embedding.

Doesn't your second *, that f is a homeomorphism onto its image f(X), automatically mean that f is injective, your first *?

 

[Landau]

Yes, you are right, the first * is superfluous. So, in short, an embedding is a homemomorphism onto its image.

 

[RedX]

So, in short, an embedding is a homemomorphism onto its image.

That makes really good intuitive sense. But for some reason the book I have defines embedding in a slightly esoteric way. Instead of topological spaces, it talks about embedding of differentiable manifolds.

First it defines an immersion. Basically, a smooth map f: M -> N between manifolds induces a map called the "differential map" f* between the vector spaces of the two manifolds:

f*: T(M) -> T(N) (explicitly, this map happens to be the Jacobian).

If f* is an injection, then f is said to be an immersion.

In addition, if f is an injection, then f is said to be an embedding.

This definition of an embedding seems weird to me, not as nice as your definition. Is there a reason why one would define an embedding in this way?

 

[Landau]

Remember, we were talking about topological spaces, i.e. spaces with as additional structure (only) a topology. My definition was of a topological embedding, which is just called an embedding if the context (namely, topology only) should be clear.

In your book it is about (smooth) manifolds: those are first of all topological spaces, but they also have extra structure, namely a smooth structure (atlas). An embedding between smooth manifolds, let us call it a smooth embedding to be sure, also has to deal with this smooth structure, and that is the immersion part.

So, a smooth embedding is a topological embedding which is at the same time an immersion (= injective differential).