Thread: Topology Proofs, Bases

1) B={(a,b)|a,b are rational numbers} is a countable base for the usual topology for the real numbers.

2) For any topologies T1 and T2 for X, if B1 is a base for T1 and B1 is a subset of T1, then T1 is a subset of T2

Def1: Bp is a local base for p in (X,T) iff p is an element of B and B is an element of T for every B an element of Bp; and when p is an ellement of G is an element of T, then there is Bg and element of Bp such that p is an element of Bg, which is a subset of G.

Def2: B is a base for the topology T iff B= the union of all Bp st p is an element of X, where each Bp is a loca base for p in the topological space (X,T).

Originally Posted by Kiefer

1) B={(a,b)|a,b are rational numbers} is a countable base for the usual topology for the real numbers.

You should prove this yourself. It's very easy by the definition of a base and if you know some basic facts about countabilitiy (rationals are countable, products of countable are countable.)

If you're going to continue in math, this is an important fact & example that you should understand well.

Having a countable base comes up often enough that it's given it's own name. "A topological space is second countable if it has a countable base." This example, exploiting that the rationals are dense in the reals to produce a countable base (generalized to ), is the "stock example" of what/why/how a base is countable.

(And the exploitability of countable dense subsets is itself a common/useful enough feature to have it's own name, "separable". Furthermore, these two ideas, separable and second countable, are often considered together, and are equivalent properties for nice spaces.)

Originally Posted by Kiefer

2) For any topologies T1 and T2 for X, if B1 is a base for T1 and B1 is a subset of T1, then T1 is a subset of T2

Are you sure you didn't make a mistake in writing down the problem?

Again, if you're going to continue in math, it might be worth investing a little time learning some basic LaTex. Your entire post was kinda difficult to read, but would've been very easy to read had it been done in LaTex.