Topology Proofs, Bases

• Sep 24th 2012, 10:34 AM
Kiefer
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).
• Sep 25th 2012, 03:54 AM
johnsomeone
Re: Topology Proofs, Bases
Quote:

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 \$\displaystyle \mathbb{R}^n\$), 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.)

Quote:

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.