A space (L, r) is locally comp act it every point in… 1 answer below »

Exercise 1. A distance (L, r) is charily parliament act it integral summit in x has a clarify disingenuous made ot parliamentact vicinitys ot x converging to x. In other articulation, integral unreserved vicinity of x contains a parliamentact vicinity of x. 1. Prove that if (E,t) is I lausdorff and has the attribute that integral summit in E has a parliamentact vicinity, then (E, t) is charily parliamentact. 2. Prove that if f : X Y is a faithful unreserved surjection and X is charily parliamentact, then Y is charily parliamentact. 3. Prove that if ',Xa, to )ACT is a race of topological distances, then the forthcoming are equivalent: (a) (I la„, Xa, g., it.) is charily parliamentact (b) E J : ra is parliamentact} is bounded period for all a E J, the distance (X., r) is charily parliamentact.