answer plz if any one question you can understand

12:51 Hill 42 a) Give an example of the Hausdorff space which is not regular. b) Let GI (2, C) be a group under multiplication of a all 2 x 2 matrices with complex entries and (did , dy ,d2) EC* . What do you think, is there exist a homeomorphism map between these two topological spaces. Justify your answer by giving a logical statement. Answer: Q.No.2. (4.5+4.5) a) A continuous real-valued function on a compact space is bounded and attains its bounds. Prove it. b) Are the spaces R, C or Z form compact spaces. If the space is not compact, how we tackle its non- compactness which convert it into compact space. Answer: Q.No.3. (4.5+4.5) a) Let F Ed be arbitrary and 4, be its path component. Then 4, is the largest path connected subset of X containing p. Prove it. b) Let X be a normal space and / : 4 -> >be a continuous, closed , onto map. Then Y is normal. Answer: Q.No.4. (4.5+4.5) a) Establish the relationship if any between local connectedness and path-connectedness. b) What is the significance of compactness and connectedness? Explain this concept by giving some real application . Is there any relationship between these two terminologies? E O