Prove that if a topology ? has a countable subbase, then it also has a countable base.
Prove that if a topology ? has a countable subbase, then it also has a countable...
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the convergent sequences in X with respect to this topology. Prove that if X is uncountable, then there is a subset S of X whose closure contains points that are not limits of the sequences in S....
(3E) Slinky Line. Consider R with the usual topology. Prove that R/N (see 3.12(e)) is Hausdorff but not first countable. (3E) Slinky Line. Consider R with the usual topology. Prove that R/N (see 3.12(e)) is Hausdorff but not first countable.
fill in the blanks. (Topology) usual discrete indiscrete finite complement Lower limit countable complement K-topology 0 0 connected path connected O compact countably compact limit point compact B-W compact sequential compact 1st countable 2nd countable LindelÖf 0 Ti T2 separable usual discrete indiscrete finite complement Lower limit countable complement K-topology 0 0 connected path connected O compact countably compact limit point compact B-W compact sequential compact 1st countable 2nd countable LindelÖf 0 Ti T2 separable
Problem 6 Suppose A and B are countable sets. Prove A × B is a countable set.
prove that the set of all algebraic number is countable
Topology Prove that if X and Y are connected topological spaces, then X x Y with the product topology is connected
"Topology" 22. Prove that any projection function is open.
2. Prove the following Theorems: (a). Prove that the real line with the standard topology is Hausdorff. (b). Prove that int(ANB) = int(A) n int(B) Y is a homeomorphism. Then if X is a (c). If X and Y are topological spaces and f: X Hausdorff space then Y is Hausdorff. (d). Theorem 4.2
Problem 1. Let A C R be a countable set. Prove that R\ A is uncountable.
Prove that in R^n with the usual topology, if a set is closed and bounded then it is compact.