Let E and F be compact sets in a metric space (X,d). Show that E U F is compact using the definition of compactness.

Q: 2) let X and y be metric spaces and T. XY Y a ContinUOus mapping Then the image of a Compact subset…

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Q: Prove that if f is a continuous mapping of a compact metric space, X, into a metric space Y, then…

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Q: (3) Let R² and R be endowed with their Euclidean metrics. (a) Show that projections p₁(x, y) = x and…

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Q: Verify that the discrete metric is actually a metric.

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Q: Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and…

A: Use definition of disconnectedness mentioned below.

Q: Consider A = [3,7), B = (7,9] as a subset of R with the usual metric  dR (x,y) = abs(x-y). Show Ac…

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Q: Exercise. Prove that every finite subset of a metric space is closed set.

A: We have to Prove that every finite subset of a metric space is closed set.

Q: (b) Show that the metric space (Q,1) is not complete. (c) Prove that the discrete metric space (M,d)…

A: Complete Metric Spaces :- A metric space (X, d) is said to be complete iff every Cauchy sequence in…