Question
i want to prove the theorm (3.10 ) please see the file
of R2 studied in Chapter 2. The topologies will, of course, be the set of all open sets turns out to be the same. Definition. Let X be a topological space with two bases B and B'. Let J and J' be the collections of all open sets corresponding to these bases. Then B is equivalent to B' if J = J'. Theorem. Let X be a topological space with two bases B and B'. Then B is equivalent to B' if and only if for each B B and each x B , there is a B' B' such that x B' B; for each B' B' and each x B', there is a B B such that x B B'. Prove Theorem 3.10. Show that the two bases for R2, the standard euclidean topology of Chapter 2 and that illustrated in Figure 3.2, are equivalent. Equivalent bases give the same collection of open sets, and so the same closed sets, connected sets, and continuous functions. We can use these cases interchangeably. A different structure on R1 is defined by using the half-open topology: an interval of the form (a, b) be considered as a
Explanation / Answer
can you repost this question ~ 12 at night india time ? i assure you i will prove this.
