Show that if a topological space has a finite number of points each of which is closed then it has the discrete topology.

Q: Show in the discrete metric space: 1. All sets are open (especially single points);

A: Let X be a non-empty set and d be discrete metric on X

Q: Is R, equipped with the finite-closed topology, Hausdorff? Provide a proof supporting your answer.

A: Given: ℝ is equipped with the finite-closed topology. Let, ℝ with this topology be Hausdorff and x,…

Q: Show with all the relevant details, albeit succinctly that every compact metric space M is complete.

A: Here we use sequential definition of compactness so that we use that in proving the completeness of…

Q: Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.

A: A set is said to be infinite countable if its elements can be counted like natural number but you…

Q: are closed.

A:

Q: A topological space is Hausdorff iff limits of all nets in it are unique-

A: This is a problem of Topology.

Q: a) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

Q: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

Q: 6. Show that a compact metric space X is locally compact.

A: The given problem is related with metric space. We have to show that a compact metric space X is…

Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

A:

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

Q: Let A be any set with the discrete topology. Show that A is normal.

A:

Q: Let X be a topological space. Prove that (a) Ø and X are closed. (b) Intersections of any number of…

A: Union of any open sets is open and intersection of two open sets is open.

Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

Q: show that (0.1)with usual topology is connected

A: Sol

Q: Prove that T is the discrete topology for X iff every point in X is an open set

A:

Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

A:

Q: Prove that every finite T,- space has the discrete topology.

A: T=(X, τ)  (X, τ) is a T1 space if and only if all points of X are closed in T.   Let X≠∅ be a set.…

Q: Exercise 2.7. Give an example of a topological space and a collection of open sets in that…

A:

Q: Exercise 2.27. Show that a set U is open in a topological space X if and only if every point of U is…

A: Let, U is an open set in a topological space X. Then, U⊂U  So, U⊂int(U) and int(U)⊂U Hence, U=int(U)

Q: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

Q: Hausdorff space.

A: Given: A set X with co-finite topology and  Hausdorff space. To Find: Under what condition a set X…

Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

A:

Q: Let (X, M, u) be a measure space wE

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

Q: Show that any infinite subset A of a discrete topological space X is not compact.

A:

Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

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

A: Topology Question

Q: if

A: R is not connected if T is the finite closed topology.

Q: Prove that the image of a continuous mapping of a compact set is compact.

A:

Q: 1. prove every 2nd countable space is Lindelof space. 2. Every Lindelof topological space induced by…

A:

Q: Question No 1 (8). Prove that the discrete topology on an uncountable set does not Satisty the…

A: Note- Let X be a topological space, then X satisfies the second axiom of countability or is second…

Q: Show that the closed ball Y in a metric space (x,d) is a closed set .Also show that if (x,d) is…

A:

Q: a) Consider the set X = {1,2,3} with the topology t = Show that (X, t) is a topological space. (0,X,…

A: As per our company guidelines we are supposed to answer only one asked question. Kindly repost…

Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

A:

Q: Show that the closed ball Y in a metric space (X, d) is a closed set. Also, show that if (X, d) is…

A:

Q: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: The intersection of any countable family of open dense-sets in a locally compact space X is dense.

A:

Q: what ordered pair is an example of a topological space that is connected but not path-connected,why?

A: We use contradiction method.

Q: (a) A discrete space with more than one point is disconnected. (b) Any trivial space is connected…

A:

Q: Let (X,τ) is a topological space and A ⊆ X. If all subsets of A are closed in X, then set A cannot…

A: Given that X,τ is a topological space. Let's prove this theorem by contradiction. So  X,τ is a…

Q: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

Q: Theorem 6.5. A space X is compact if and only if every collection of closed sets with the finite…

A:

Q: Every closed subset of a connected metric space is connected. True False

A:

Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

A:

Q: Let X be ametric space. If A and B are each dense open subsets of X then AOB is dense open subset of…

A: First we have shown the intersection is open. Then we use the property that if a set intersect every…

Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then* OX…

A: Gievn that Let T be the finite closed topology and be also the discrete topology on a non empty set…

Q: Show that the closure of a connected space is connected.

A: