# Definition of a topological property

"A topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets." (I copied it from Wikipedia)

Now my question is: What is the definition of a topological property ? Of course you can define it as wiki defines it. But I am more concerned about the the part of wiki's "definition" which says that "Informally, a topological property is a property of the space that can be expressed using open sets."

Is there a definition of a topological property that says which well formed formulas are well formed formulas of topological properties and which are not ?

Because of what I read in wikipedia, I was expecting to see a definition of a topological property that talks about the internal structure of the well formed formula of the property. Then, I also expected that there was a theorem that says that if $(X_1,T_1),(X_2,T_2)$ are any two homeomorphic topological spaces and the well formed formula $\phi(X,T)$ is a topological property, then:

$\phi(X_1,T_1)$ iff $\phi(X_2,T_2)$

Is there such a definition and such a theorem ?

Such a definition and such a theorem will enable one to spot many topological properties easily.

Here is a similar question: Can you characterize all properties of topological spaces which are preserved by homeomorphisms

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"Closure" only makes sense in a parent space. Homeomorphism is about the spaces themselves, separate from any superset they might have been constructed from. –  Thomas Andrews Nov 27 '12 at 14:49
Here's how you can "fix" the definition from Wikipedia: a topological property of a space $X$ is that which can be expressed using open sets in $X$ itself. In your case, closure of the unit ball uses open sets in the whole $\mathbb{R}^2$, not open subsets of the unit ball itself. –  Dan Shved Nov 27 '12 at 14:53
A topological property is one that is preserved by homeomorphism :) –  Thomas Andrews Nov 27 '12 at 14:59
@Graphth: My understanding is that the OP is looking for a syntactic criterion such that all formulas that fit within the restricted syntax will define topological properties, and such that a usefully large variety of topological properties can be expressed within the restricted syntax. Some appropriate type discipline could undoubtedly work, but I'm not sure whether it is syntactic enough for the OP. –  Henning Makholm Nov 27 '12 at 15:14
The properties that are considered "topological" are fairly complex. You are going to have a tough time coming up with a formal language that will establish properties that are "topological" that will include homotopy and homology and other things that use maps to other categories. That said, I think you see the error of your interpretation of the Wikipedia page. –  Thomas Andrews Nov 27 '12 at 15:49

We can formalise topology using only the language of set theory. [For instance, a topological space is a pair $\langle X, \tau \rangle$ where $X$ and $\tau$ are sets satisfying various properties, and we can define a homeomorphism $\langle X, \tau \rangle \to \langle Y, \sigma \rangle$ as a function $X \to Y$ (which is itself a set) satisfying some conditions, etc. All this can be formalised.]

So we can define a unary predicate $\text{TS}$ defined by $$\forall x[\text{TS}(x) \leftrightarrow x\ \text{is a topological space}]$$ where '$x\ \text{is a topological space}$' is shorthand for... \begin{align}\exists X \exists \tau( \hspace{53pt}\\ x= \langle X, \tau \rangle \wedge &\tau \subseteq \mathcal{P}(X) \wedge \varnothing \in \tau \wedge X \in \tau\\ \wedge & \forall U\ \forall V\ [U \in \tau \wedge V \in \tau \to U \cap V \in \tau]\\ \wedge & \forall A\ [A \subseteq \tau \to \bigcup A \in \tau]\\ ) \hspace{78pt} \end{align}

Now suppose $\phi$ is a formula with one free variable, $x$ say. Then $\phi$ is a topological property (i.e. is preserved under homeomorphism) if $$\forall x [\text{TS}(x) \wedge \phi(x) \rightarrow \forall y[\text{TS}(y) \wedge x \cong y \rightarrow \phi(y)]]$$ That is, if $\phi$ holds for any space $x$ then for any space $y$ homeomorphic to $x$, $\phi$ holds for $y$.

Here I've used $x \cong y$ as shorthand for the formula expressing that $x=\langle X, \tau \rangle$ and $y=\langle Y, \sigma \rangle$ are homeomorphic.

Is this what you were after?

Frankly, I don't see how it's any more enlightening to put yourself through all this than it is to just say "a topological property is one that is preserved by homeomorphism", as so succinctly put by Thomas Andrews in the comments.

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I will edit my question to say what I had in mind. –  Amr Nov 27 '12 at 16:26
@Amr: Having seen your edit, I'm still a bit puzzled; what's the theorem you're talking about? If a topological property is defined to be a property which is invariant under homeomorphism, then the theorem stating that a topological property is invariant under homeomorphism is a tautology! [But I do think my response answers your question - if it doesn't, please let me know what else you want to know.] –  Clive Newstead Nov 27 '12 at 16:46
The theorem puts conditions on the the structure of the well formed formula $\phi$ (positions of quantifiers ...) and the conclusion is that homeomorphisms preserve the property $\phi$ –  Amr Nov 27 '12 at 16:51
@Amr: But any first-order formula $\phi$, i.e. a property of sets, could potentially be a topological property. For instance, any property of sets preserved under bijections of sets is a topological property under this definition; as is any theorem of the predicate calculus. I very much doubt such a theorem on the structure of $\phi$ exists; if it did then topology would become trivial! –  Clive Newstead Nov 27 '12 at 16:53
Here is another question (harder than the first one but hopefully clearer): can you characterize all properties of topological spaces that are preserved by homeomorphisms –  Amr Nov 27 '12 at 17:25
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There is a fundamental misunderstanding about what topology is actually capabable of doing. An invariant is a way to calculate whether two space could be homeomorphic. Take the Euler characteristic. Just because given two spaces that have the same Euler characteristic doesn't mean that they are homeomorphic. However, if two space don't have the same Euler characteristic then the are definitely not homeomorphic. There is very much a zen thing that happens in topology. To answer your question more directly you could make an exhaustive list of invariants but you would not know if it was complete.

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