Homeomorphisms in Topology
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Homeomorphisms in Topology
Hey everyone, another question relating to Topology.
So, everyone seems to agree that, given two topological spaces (X, T1) and (Y, T2), a homeomorphism between the two spaces is a bicontinuous function between X and Y. That much makes sense, but my question is, why doesn't the definition of the homeomorphism make any mention of the topologies T1 and T2? Surely they must be important somehow; is there something implicit in the definition that relates them somehow, or are we really able to say that if X and Y are related by a bijection, they'll have the same topological properties regardless of the topologies that we equip them with? Basically, how would changing T1 and T2 change the homeomorphism relationship between X and Y?
I'm sure my question sounds rather confused, but I'd appreciate any help.
So, everyone seems to agree that, given two topological spaces (X, T1) and (Y, T2), a homeomorphism between the two spaces is a bicontinuous function between X and Y. That much makes sense, but my question is, why doesn't the definition of the homeomorphism make any mention of the topologies T1 and T2? Surely they must be important somehow; is there something implicit in the definition that relates them somehow, or are we really able to say that if X and Y are related by a bijection, they'll have the same topological properties regardless of the topologies that we equip them with? Basically, how would changing T1 and T2 change the homeomorphism relationship between X and Y?
I'm sure my question sounds rather confused, but I'd appreciate any help.
 MartianInvader
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Re: Homeomorphisms in Topology
The important word there is "bicontinuous". Not just any bijection is a homeomorphism. For example, I could take the Cantor set and find a bijection with the real numbers (both are uncountable), but I could never make it continuous both ways. This is because the two have very different topologies.
Being bicontiunuous depends very heavily on the topologies of both spaces.
Being bicontiunuous depends very heavily on the topologies of both spaces.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
Re: Homeomorphisms in Topology
The definition does mention the topologies on X and Y. It's in the word "bicontinuous". Whether a function X>Y is continuous depends on the topologies on X and Y.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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Re: Homeomorphisms in Topology
I think I just wasn't thinking about continuity in the topological sense; continuity makes reference to the open sets of a topological space, which are of course determined by the topology on that space.
I think I can figure it out from here, thanks for the reply
EDIT: Thanks atonfire, I figured this out just as you posted it.
I think I can figure it out from here, thanks for the reply
EDIT: Thanks atonfire, I figured this out just as you posted it.
Re: Homeomorphisms in Topology
Captain_Thunder wrote:they'll have the same topological properties regardless of the topologies that we equip them with?
I think you're missing something fundamental here. A topology completely specifies the topological properties of its underlying space. The underlying space is just a convenient metaphor; it is not actually necessary.
 Incompetent
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Re: Homeomorphisms in Topology
Captain_Thunder wrote:I think I just wasn't thinking about continuity in the topological sense; continuity makes reference to the open sets of a topological space, which are of course determined by the topology on that space.
The topology doesn't just 'determine' the open sets; it 'is' the open sets. A topology on a set is just a collection of subsets that is closed under arbitrary unions and finite intersections, and which contains the empty set and the whole set. It has no more structure than this. Every topological concept must be definable *entirely* in terms of basic set/logical operations and whether or not certain sets are members of the topology, or it can't be purely topological.
The hard thing to get used to about topological spaces is just how little structure is imposed by the axioms. For instance, the topology arising from an arbitrary metric space is already much more special and wellbehaved than an arbitrary topology.
 jestingrabbit
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Re: Homeomorphisms in Topology
t0rajir0u wrote:Captain_Thunder wrote:they'll have the same topological properties regardless of the topologies that we equip them with?
I think you're missing something fundamental here. A topology completely specifies the topological properties of its underlying space. The underlying space is just a convenient metaphor; it is not actually necessary.
As is pointed out at that link, pointset topology and pointless topology yield different theories. You can easily study one whilst not studying the other.
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Re: Homeomorphisms in Topology
Point taken. I guess what I'm really trying to get at is that Stone duality is an important point of view to keep in mind philosophically, even if it isn't with respect to all topological spaces.
Re: Homeomorphisms in Topology
Incompetent wrote:The topology doesn't just 'determine' the open sets; it 'is' the open sets.
Only if you define it that way. You can also define it as the set of closed sets, or as the closure function, or as the interior function. For comparison, are the reals equivalence classes of Cauchy sequences of rationals, or are they Dedekind cuts of rationals? Is an ordered pair a set of the form {a,{a,b}}, or a function on the set {0,1}?
It's true that there is often a conventional definition for a type of object, but there are usually several other equivalent ones, and once you prove a few basic properties, which definition you're using hardly matters. So I think it's better to say that the topology determines the open sets. (And is determined by them.)
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
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Re: Homeomorphisms in Topology
antonfire wrote:Only if you define it that way. You can also define it as the set of closed sets, or as the closure function, or as the interior function. For comparison, are the reals equivalence classes of Cauchy sequences of rationals, or are they Dedekind cuts of rationals? Is an ordered pair a set of the form {a,{a,b}}, or a function on the set {0,1}?
It's true that there is often a conventional definition for a type of object, but there are usually several other equivalent ones, and once you prove a few basic properties, which definition you're using hardly matters. So I think it's better to say that the topology determines the open sets. (And is determined by them.)
My objection is to the connotations, not so much a matter of formal correctness.
The alternatives you give for defining a topology are trivially equivalent, and it's ok to switch between definitions like this one a canonical equivalence has been established. Nevertheless, I disagree with your last sentence  I think talking about open sets being 'determined by the topology' is misleading. It's a bit like saying 'edges are determined by the graph' or 'multiplication is determined by the group structure'  it's tautological, and although tautologies are formally correct, they can easily lead to confusion when someone reading the sentence looks in vain for some substantive content in it. In this case, one might think that a topology is an essentially more complex beast than a set of open sets, with the former possessing also 'closed sets', an 'interior function', a set of 'continuous maps' to/from any other given space, and so on, and that the right way to obtain the open sets is to study this complicated entity and use general properties of its structure to 'determine' whether or not a given subset is open.
 Yakk
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Re: Homeomorphisms in Topology
So I remember a construction of Topology that was based off of the set of functions that where continuous (on the set to itself?) And from that, you would derive the concept of 'openness'.
I cannot remember where I saw it.
I cannot remember where I saw it.
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Re: Homeomorphisms in Topology
In the modern treatment, openness dictates the notion of continuity, not the other way around. It's certainly possible to take, say, the coarsest topology such that some space of functions is continuous (this is how the product topology is defined), but usually it will be harder to describe the continuous functions than to describe the open sets.
Re: Homeomorphisms in Topology
Here is a definition of homeomorphic spaces which might appeal to you more (because it makes references to the open sets explicit):
Two topological spaces are said to be homeomorphic when there is a bijection between their sets which extends to a bijection between their topologies. That is if [imath](X,\tau_1)[/imath] and [imath](Y,\tau_2)[/imath] are spaces, they are homeomorphic iff there is a one to one and onto function [imath]f: X \rightarrow Y[/imath] such that the map [imath]P_f: P(X) \rightarrow P(Y)[/imath] defined by [imath]P_f(U) = \{ y \in Y  y = f(u) \text{ for some $u$ in $U$} \}[/imath] restricts to a bijection between the topologies,[imath]P_f_{\tau_1}: \tau_1 \rightarrow \tau_2[/imath].
Basically what this definition says is that we have a dictionary which allows us to translate points in [imath]X[/imath] to points of [imath]Y[/imath] in such a way that the open sets of [imath]\tau_1[/imath] get renamed as open sets of [imath]\tau_2[/imath] and vice versa.
Important Exercises:
1. Check that my definition of a continuous function is the same as the standard one.
2. Show that bicontinuity is necessary, i.e. show that you can have a continuous bijection [imath]f: X \rightarrow Y[/imath] without having [imath]X[/imath] and [imath]Y[/imath] homeomorphic.
Two topological spaces are said to be homeomorphic when there is a bijection between their sets which extends to a bijection between their topologies. That is if [imath](X,\tau_1)[/imath] and [imath](Y,\tau_2)[/imath] are spaces, they are homeomorphic iff there is a one to one and onto function [imath]f: X \rightarrow Y[/imath] such that the map [imath]P_f: P(X) \rightarrow P(Y)[/imath] defined by [imath]P_f(U) = \{ y \in Y  y = f(u) \text{ for some $u$ in $U$} \}[/imath] restricts to a bijection between the topologies,[imath]P_f_{\tau_1}: \tau_1 \rightarrow \tau_2[/imath].
Basically what this definition says is that we have a dictionary which allows us to translate points in [imath]X[/imath] to points of [imath]Y[/imath] in such a way that the open sets of [imath]\tau_1[/imath] get renamed as open sets of [imath]\tau_2[/imath] and vice versa.
Important Exercises:
1. Check that my definition of a continuous function is the same as the standard one.
2. Show that bicontinuity is necessary, i.e. show that you can have a continuous bijection [imath]f: X \rightarrow Y[/imath] without having [imath]X[/imath] and [imath]Y[/imath] homeomorphic.
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