37 and 38 please??? Let X be a topological space, let p epsilon X, let Np be the

Question

37 and 38 please???

Let X be a topological space, let p epsilon X, let Np be the neighborhood system at p and left Bp be a local base at p. Show that every neighborhood of p contains a member of the local base at p; i.e. for every N epsilon Np, ?G epsilon Bp for which G sub N. Show that if a point p has a finite local base Bp then it also has a local base consisting of exactly one set. Consider the upper limit topology T on the real line R which has as a base the class of open-closed intervals (a. b]. Determine whether or not each of the following sequences converges: (1, 1/2, 1/2.), (-1,-1/2,-1/2,), (-1, 1/2, -1/2, 1/2,). Let T be the topology on the real line R generated by the class of all closed intervals [a b] where a and b are rational (see Problem 30). Determine whether or not each of the following sequences converges: (2 + 1/2. 2+ 1/2, 2+ 1/2,), (root2 + 1/2, root2 + 1/2, root2 + ¼,). Determine the closure of each of the following subsets of R: (2,4), (root2,5), (-3,pi), (d) A = (1, 1/2, 1/2,}. Show that any finite subset of R is T-closed. Let be a subbase for a topological space X and let p epsilon X. Show by a counterexample that the class = {S epsilon : p epsilon S) need not be a local base at p. Show that finite intersections of members of do form a local base at p. Show that a sequence (an) in X converges to p if and only if every S epsilon contains all except a finite number of the terms of the sequence.

Explanation / Answer

1)

(-1, 0] does not contain any term of the sequence, therefore it does not converge.

2) For any open basic set (a, b] containing 0 (i.e. for which a < 0 <= b) there exist n0 in N such that a < -1/n0.

therefore n>n0 implies 1/n in (a, b]. Thus it converges.

3) it contains non-convergent sequence

therefore it doesnot converge.

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