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Define X = {0, 1} and T = {∅, {0}, X} . (a) Is X with...

Define X = {0, 1} and T = {∅, {0}, X} .

(a) Is X with topology T connected? (Hint: Use the clopen definition.)

(b) Is X with topology T path-connected? (Hint: Construct continuous map f ∶ R → X. One way is to ensure f −1({0}) = (−∞, 0). Once you have f, consider f([a, b])—like f([−1, 1]) if you take my suggestion. Use the definition of path connected. )

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Answer #1

as inverse image of open set is open so f is contineous

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