A question about subspace topolgy

[ShahidNawaz]

Let (X, τ) be a topolgical space. Let f: τ→R be a map that assigns real values to the elements of τ. Let (A,τ_A) be subspace of (X,τ). Let g:τ_A→R be another map that assigns real values to the element of subspace topology. My question is how the function g is related with function f given that f is a function on the parant topolgy and g is a function on the daughter topology. In the particular case in which I'm interested τ_A is a subset of τ.

 

[micromass]

You have a function that assigns values to open sets?? Like a measure??

Well, to answer your question: f and g have no connection what-so-ever. You just chose them as arbitrary functions without any conditions. So I can't see any connection between the functions...

What kind of function were you hoping for?

 

[ShahidNawaz]

Thanks for the quick reply. Yes it is a kind of measure. Let me further clarify my problem. Let T be a topology on X. Now assign real values to each open set of X by f:T->R. Next I remove the whole neighborhood system (N_x) of a prticular point x from T, i.e. T\N_x. Next I have shown that T\N_x is a subspace topology of X .Please find details of my paper in http://arxiv.org/abs/1006.2481 . Now since T\N_x call it T_A is also a topology on set A subset of X it means that we can assign reals values to T_A in the same manner say g: T_A->R. What I'm wondering g might be related with f by some sort of annihilation operator i.e. g = Df where D is an annihilation operator that destroy some elements of f. This is my tentive approach to this problem. But I'm not sure how to get g given f.

 

[micromass]

Allow me to make a general remark on your paper: you don't want to deal with topologies! A topology seems to be the entirely wrong framework for what you want to do!

You should be looking at $\sigma$-algebra's, Boolean algebra's, distributive lattices, orthomodular lattices, etc. These are the kinds of structures that are made for what you want to do! I don't really see why a topology should be more convenient than all these structures...

Instead of a neighborhood system, you want to deal with filters and ideals on the lattice. You would probably even want to work with congruence relations and quotient lattices.

As for your question: it is not clear to me what the relation between f and g is that you want. I don't even know what you want to do with f and g. Again, your question is troubles because working with a topology is not the best thing to do.

 

[ShahidNawaz]

I agree that's the standard way to approach probability theory. I was however interested in some weaker form of probability theory. Thanks again for the quick response.

 

[zhentil]

In short, there is no way this will work.

In long, you want a commutative diagram where g is given by the composition of f with an annihilation operator. This means:
1) You want to be able to assign a number to an element E\A. This will not in general be in your domain.
2) You want f(D(E\A)) to be zero and you want f(D(E intersect A)) to be g(E intersect A). This implies that your function is defined on non-open subsets AND that it's additive on disjoint unions.

This, of course, is the only way to recover g from f. Otherwise, they have nothing to do with each other. So, in short, the only way to get what you want is to take f to be a measure and g to be an induced measure.

 

[ShahidNawaz]

Yesterday I was all day working on commutative diagrams to make a connection between g and f. My guess was either g is marginalization of f or g is conditioning of f or something between it. It turns out that g is induced measure of f that is g corresponds to conditional probability provided the spaces I'm considering are probability spaces.