3. A space is totally disconnected if and only if the only connected sets are the sets of a single point. Prove that if X has the discrete topology then it is totally disconnected, is the converse implication true?

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A: As per our guidelines we are supposed to answer only first part. Kindly repost other parts in the…

Q: Let Gbe an open set in R". Two points x, Y e G are said to be equivalent if they can be joined by a…

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Q: are closed.

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Q: show that any infinite discrete topological space is not compact

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Q: 17. Prove: S' is a closed set.

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A: WE GIVE A DIRECT PROOF. First, we prove that set A is closed then A is bounded.

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A: the solution is given below

Q: Let Gbe an open set in R". Two points x, y e G are said to be equivalent if they can be joined by a…

A: Since every open subset of Rn can be written of countable union of disjoint open balls and each ball…

Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

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Q: 4. Let X be any set and 7 be the countable complement topology on X. b) If X is uncountable, then…

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Q: topological space a

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Q: Any connected component of Xis a closed set and X is disjoint Union of ts connected components,

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Q: Prove that a closed convex set in R" has a finite number of extreme points if and only it is the…

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Q: Theorem. Show that Closed subsets of compact sets are compact.  Need handwriting solution only.

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Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: if

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A: This is a question of topology.

Q: Show that any subset of bounded set is bounded

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Q: Let Gbe an open set in R". Two points x, y e G are said to be equivalent if they can be joined by a…

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A: Topology question.

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Q: Show that topologically completeness is weak hereditary property.

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