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Prove that the subspace.of a complete metric is complete if and only if it is closed


Give an example of two metric on R2 which are equivalent ,substantiate your claim


Prove or disaprove ,The closure of an open vall in a metric space.is the corresponding closed ball
Determine the closed subsets of a.set X with the discrete metric
Give an example of a locally connected space which is not connected substantiate your claim.
Prove that a subspace of a real line with the usual topology is connected if and only if it is an interval
Prove that any open continuous image of a locally connected space is locally connected
Prove that a countable product of connected space is connected
Prove that any set with the cofinite topology is compact
Prove that any collection of subsets of a set X is a sub base for some topology on X
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