Lecture - 8A: Subbasis of Topology

Transcription

1 Lecture - 8A: Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014

2 Outline 1 Introduction 2 3 4

3 Introduction I As we know that topology generated by a basis B may be described as the collection of arbitrary unions of elements of B, but what happens if we start with a given collection of sets and take finite intersection of them as well as arbitrary unions? This question leads to the notion of a subbasis for a topology.

4 I Definition () A subbasis S for a topology on X is a collection of subsets of X whose union equals to X. Or Let X be a set. Then a sub-basis is a collection S such that S S S = X. Alternatively we can say that Let X be a topological space with topology τ. Then S τ is a sub-basis for X if for each open set U and each x U, there exist finitely many S 1,..., S n S such that x S 1..., S n U.

5 II Definition (Topology generated by Subbasis) The topology generated by the subbasis S is defined to be the collection τ of all unions of finite intersections of elements of S. Or The topology generated by S is defined by the rule: U X is open if for each x U we can find S 1,..., S n such that x S 1..., S n U.

6 I Let X = {a, b, c, d, e} and let A = {{a, b, c}, {c, d}, {d, e}}, then the topology on X generated by A can be obtained as follows: First compute the class of B of all finite intersection of sets in A : B = {X, {a, b, c}, {c, d}, {d, e}, {c}, {d}, φ} (Note that X B, since by definition X is the empty intersection of members of A.) Taking unions of members of B gives the class τ = {X, {a, b, c}, {c, d}, {d, e}, {c}, {d}, φ, {a, b, c, d}, {c, d, e}} which is the topology on X generated by A.

7 II A basic example of a sub-basis is any basis.

8 III Let S = {(a, ): a R} {(, b): b R} is the set of all infinite rays in R. Then S is clearly not a basis for the standard topology on R. For example: (0, 1) is open in R, but I can t find an infinite ray containing 1/2 which is contained in (0, 1). But S is a sub-basis, for (a, b) = (a, ) (, b), so if U is an open set containing the point x, and x (a, b) U, we have x (a, ) (, b) U.

9 Difference between basis and subbasis in a topology I Question: What is/are the difference between basis and subbasis in a topology?

10 Difference between basis and subbasis in a topology II 1 Bases and subbases generate a topology in different ways. 2 Every open set is a union of basis elements. 3 Every open set is a union of finite intersections of subbasis elements. For this reason, we can take a smaller set as our subbasis, and that sometimes makes proving things about the topology easier. We get to use a smaller set for our proof, but we pay for it; with a subbasis we need to worry about finite intersections, whereas we did not have to worry about that in the case of a basis.

11 Difference between basis and subbasis in a topology III Consider S = {{0, 1}, {0, 2}}. What is the topological space τ(s) generated by S? By definition, S will then be a subbasis of τ(s). Well, we want to all requirements to hold true and find that τ(s) = {φ, {0}, {0, 1}, {0, 2}, {0, 1, 2}} (check this!) Is S a basis? No, because you cannot write {0} as a union of any elements in S. So you see that subbasis and basis are two different notions, even for a very basic example.