Different induced topologies

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In summary, the mapping f from \mathbb{R} to S^1 does not give the usual topology in S^1. The topology in S^1 is dependent on the mapping f. Furthermore, there is no way to get the usual topology in S^1 using the mapping f only.
  • #1
jostpuur
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If Y is a topological space, and f:X->Y is some mapping, we can always give X the induced topology, which consists of the preimages of the open sets in Y. However, this doesn't seem to be a good way if f is not injective. Aren't the open sets in X being left unnecessarily big?

Suppose we have the usual topology in S^1 and the natural mapping [itex]f:\mathbb{R}\to S^1[/itex], [itex]f(x)=(\cos(x),\sin(x))[/itex]. Is the induced topology from S^1 the usual topology of R? It doesn't look like that to me. All the open sets are periodic.

I though I could define a more reasonable topology to X like this: [itex]V\subset X[/itex] is open [itex]\Longleftrightarrow[/itex] [itex]f(V)\subset Y[/itex] is open. Now also smaller sets in X can be open. But I noticed that I don't know how to prove that the finite intersection of such "open sets" would still be open, because [itex]f(\cap V_i)=\cap f(V_i)[/itex] doesn't work, and now I'm not sure if this way of trying to define topology works at all.

So... How do you get the topology from Y to the X so that open sets are not forced to be too big?
 
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  • #2
jostpuur said:
If
Suppose we have the usual topology in S^1 and the natural mapping [itex]f:\mathbb{R}\to S^1[/itex], [itex]f(x)=(\cos(x),\sin(x))[/itex]. Is the induced topology from S^1 the usual topology of R? It doesn't look like that to me. All the open sets are periodic.

Must it be R? No.

Try looking at the Quotient Topology section in your textbook.
 
  • #3
JasonRox said:
Try looking at the Quotient Topology section in your textbook.

If I have topology in X, and an equivalence relation there, then I can define a quotient topology in [itex]X/\sim[/itex], but [itex]X/\sim[/itex] is smaller set than X. I don't see how this helps me here, because I'm now trying to get a topology from a smaller set to a bigger one. More precisely, I want a topology from Y to X, when f:X->Y is not injective.

Or did you mean, that we get the topology from [itex]X/\sim[/itex] to X?
 
  • #4
A quick thought! If I say that the collection of sets [itex]\{V\subset X\;|\;f(V)\subset Y\;\textrm{open}\}[/itex] is the subbasis of the topology in X, I guess then everything is fine?

No! That is not what I want. The topology becomes well defined, but it is not of the correct kind. For example in the case of [itex]f:\mathbb{R}\to S^1[/itex], a set [itex]]0,1/2]\cup [1/2+2\pi, 1+2\pi[[/itex] would become open.
 
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  • #5
jostpuur said:
I hope you found my monograph entertaining.

Makes me think, so I enjoy it.

I see what you're saying now. But what you did doesn't seem to go anywhere because you're letting a set be a subbasis. Which really doesn't do anything special because it's obviously going to create some kind of topology.

You might now want to ask yourself the following questions...

How does this topology on X dependent on the map f?

Is it uniquely determined by f?

What kind of structure might it have depending on Y?
 
  • #6
JasonRox said:
jostpuur said:
I hope you found my monograph entertaining.
Makes me think, so I enjoy it.

And now it continues, because I edited this temporary ending away.

I have two different ways of defining a topology to X, out of topology of Y and a mapping f:X->Y. One is the usual inducing, and the other one goes through the subbasis. At the moment I have only one requirement for the topology of X. I want to it to be a kind of topology, that if I get the topology from S^1=Y to R=X, I get the usual topology of R. These two ways give something else, so I don't like them.
 
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  • #7
I asked about this from one mathematician today, and got an answer. If we have the mapping [itex]f:\mathbb{R}\to S^1[/itex], [itex]f(t)=(\cos(t),\sin(t))[/itex], and the usual topology in [itex]S^1[/itex], there's no way of getting the usual topology to the [itex]\mathbb{R}[/itex] honestly with this mapping only.

Consider for example a sequence 1, 1/2, 1/3, ... Is it supposed to converge towards 0 or towards [itex]2\pi[/itex]? There's no way of deciding it using the f only.
 

1. What is an induced topology?

An induced topology is a way of defining a topology on a subset of a larger topological space. It is created by taking the collection of all subsets of the larger space that intersect with the subset in question, and considering these subsets as open sets. This results in a topology that is unique to the subset and may differ from the topology of the larger space.

2. What is the difference between an induced topology and a subspace topology?

While an induced topology and a subspace topology both involve defining a topology on a subset of a larger space, they differ in how they do so. An induced topology is defined by considering all subsets of the larger space that intersect with the subset in question, while a subspace topology is defined by considering all subsets of the subset itself. This can result in different open sets and therefore different topologies.

3. How is an induced topology related to continuity?

An induced topology is closely related to continuity, as it is used to define continuity for functions between topological spaces. Specifically, a function is continuous if and only if the preimage of every open set in the codomain is open in the induced topology on the domain.

4. Can an induced topology be induced by more than one larger space?

No, an induced topology is unique to the larger space it is induced from. This is because the open sets in an induced topology are defined based on the larger space, so changing the larger space would result in different open sets and therefore a different topology.

5. How are induced topologies used in practical applications?

Induced topologies have various applications in fields such as physics, engineering, and computer science. They are used to define topological spaces on subsets of larger spaces that have specific properties or behaviors, and are also useful for studying continuous functions between spaces. For example, induced topologies are used in the study of manifolds and in the design of algorithms for optimization problems.

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