Thread: Another Topologic space question!

Hi. Thanks for reading... I'm having no progress with this one.
Let X be a set, and S, and subset of P(X). Meaning, S is a set of subsets of X. (P(X) = all the subsets of X)
I need to prove that there exists a minimal topology, marked "T(S)", where S is a subset of T(S), and if there's another T' (=topology) of which S is a subset of => T(S) is a subset of T' (That's the idea of minimal - it's written better in mathematics but I don't know yet how to write it here)

Hope I explained it correcty....

I was also given a hint: try to immitate the definition of a closure of a group in a topological space. The definition I got is: cl(A)= intersection {F | A is a subset of F, F is closed}. We also proved later on that cl(A) = unification (A, A'), and cl(A) = unification (A, d(A)), where d(A) is the surface of A.


Thank you very much for reading, and even more if you dare to help!
Tomer.

Originally Posted by aurora

Hi. Thanks for reading... I'm having no progress with this one.
Let X be a set, and S, and subset of P(X). Meaning, S is a set of subsets of X. (P(X) = all the subsets of X)
I need to prove that there exists a minimal topology, marked "T(S)", where S is a subset of T(S), and if there's another T' (=topology) of which S is a subset of => T(S) is a subset of T' (That's the idea of minimal - it's written better in mathematics but I don't know yet how to write it here)

Hope I explained it correcty....

I was also given a hint: try to immitate the definition of a closure of a group in a topological space. The definition I got is: cl(A)= intersection {F | A is a subset of F, F is closed}. We also proved later on that cl(A) = unification (A, A'), and cl(A) = unification (A, d(A)), where d(A) is the surface of A.


Thank you very much for reading, and even more if you dare to help!
Tomer.

A topology on X is a subset of P(X) that is closed under finite intersections and arbitrary unions (and contains the empty set and the whole space). There are two ways of defining the minimal topology containing S.

Method 1 (the top-down method). Consider the collection of all topologies on X that contain S. This collection is nonempty (because it contains the discrete topology P(X)). It is easy to check that any intersection of topologies on X is again a topology. So define the minimal topology on S as the intersection of all topologies on X that contain S.

Method 2 (the bottom-up method). Let T'(S) be the set of all finite intersections of elements of S (together with the empty set, if necessary), and let T"(S) be the set of all unions of elements of T'(S) (together with X, if necessary). Then check that T"(S) is a topology. It's fairly obvious that it must be the smallest topology containing S.

Opalg, bless you! You're my saviour!