SHOW that the continuous image Path - connected space is path Connected.
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- Suppose metric space X is not path-wise connected then X is not connected. True FalseIs the boundary of a path-connected set in R^2 with the usual topology path-connected?Explain with an example of a topological space which is singleton that is open but not close and another one that is close but not open
- Provide a direct proof that a sequentially compact set must be both closed and bounded without taking a detour through the concept of compactness.Define what it means for a topological space to be completely regular,Is the boundary of a path-connected set in R^2 with the usual topology path-connected? Please be as clear as possible showing all the steps and giving definitions if necessary. Thank you a lot.