Let X and Y be two discrete spaces, then * X is never homeomorphic to Y X is homomorphic to Y if and only if X and Y are both infinite X is homeomorphic to Y if and only if X and Y have the same cardinality X is homeomorphic to Y if and only if X and Y are both finite

Q: (a) The intersection of any collection of closed sets in X is closed (in X). (b) The union of any…

A:

Q: 14) A subset of topological space X is dense in X iffA a) intersect every set c) intersect every…

A:

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

A:

Q: 3. Let (X, d) be a metric space and x1, x2, ... , Xn, ... a sequenc

A:

Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

Q: 3.1 Show that an infinite intersection of closed sets E k = 1,2,3,., in a metric space (SA) is a…

A:

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: In an indiscrete space X * O None of the choices O the only dense subset of X is X every subset of X…

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

A:

Q: Suppose that X and Y are topological spaces and f: X- Y. Show that f is continuous iff for each…

A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

Q: 3. Let (X, 7) be a Hausdorff space. If X is finite, then show that 7 = P(X), where P(X) is a power…

A:

Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Two discrete spaces are homeomorphic if there exists a bijection between them and it is possible if…

Q: In a discrete space X * the only dense subset of X is X every subset of X is dense in X None of the…

A: In topology , a subset A of a topological space X is called dense (in X) if every point x in X…

Q: 6. Let (X,T) be a topological space such that every subset of X is closed, then a. (X,T) is an…

A:

Q: 3.1 Show that an infinite intersection of closed sets F, k = 1,2,3, . (S, d) is a closed set. in a…

A:

Q: Let (X,T) be a topological space Property C=P.C.  A subset A of x has P.C If it's subset of the…

A:

Q: Let X and Y be two discrete spaces, then * O X is never homeomorphic to Y X is homomorphic to Y if…

A: option (c) is correct.

Q: 3.1 Show that an infinite intersection of closed sets F, k = 1,2, 3, ..., in a metric (S, d) is a…

A: Let {Fk}  be an arbitrary collection of closed sets . . 1  i.e. for k = 1, 2, 3, . . . . . . . . To…

Q: Let X and Y be two discrete spaces, then! X is never homeomorphic to Y X is homeomorphic to Y if and…

A:

Q: If X is a normed space then the open ball and closed ball is a Example: convex set

A:

Q: Construct examples to show that if X and Y are topological spaces with G acting on them such that…

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: Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point.

A: We have given that, X is Compact space.

Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

Q: 4. Let (X, d) be a metric space and xnx where each x, e X and x e X. Let A be the subset of X which…

A:

Q: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

A:

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 E₁ and E₂ be open dense subsets of a topological space X. (a) Show that E₁ E2 is dense in X. (b)…

A:

Q: 3.1 Show that an infinite intersection of closed sets Fa, k = 1, 2,3,.., in a metric space (S, d) is…

A: The given is related to Metric spaces. The solution is explained below.

Q: 3. ** Let X, Y be finite sets, and f is a function XY. What can you say about the cardinality if X,…

A: 3. Given X, Y are finite sets and f:X→Y is a function   To Comment about cardinality of X and Y if…

Q: in the discrete metric space: 1. All collections are clo

A: Let (X, d) be a metric space. Consider a closed ball Brx0 with both center x0 ∈ X and radius r >…

Q: 3. Let K,and K, be compact subsets of a metric space X. Show K, U K2 is compact

A:

Q: Let X and Y be two discrete spaces, then * X is homomorphic to Y if and only if X and Y are both…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

Q: In a discrete space X * O the only dense subset of X is X O every subset of X is dense in X None of…

A: Dear Bartleby student according to our guidelines we can answer only three subparts, or first…

Q: Let X and Y be two discrete spaces, then* O x is never homeomorphic to Y O X is homomorphic to Y if…

A: As two discrete spaces are homeomorphic iff they have same cardinality .

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

A:

Q: 6. Let (X,T) be a topological space such that every subset of X is elosed, then a. (X,T) is an…

A: Our guidelines we are supposed to answer only one question. Kindly repost other question as the next…

Q: If T1 and T2 are two topologies on the same space X, then * T1NT2 and T1UT2 are both topologies on X…

A:

Q: Let ||.|| and ||..|| be two norms defined on space X.Show that ||.|| and ||..|| are equivalent if…

A:

Q: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

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

A:

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

A: X is finite

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

A:

Q: Let X and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the…

A: X and Y are two discrete spaces then X and Y   The topology is  discrete then for x∈X  is an open…

Q: A discrete metric space X is separable if and only if X is countable

A: Given: To prove: A discrete metric space X is separable if and only if X is countable

Q: space. connected subsets of X such that C; is non-empty, then C = C; is also

A:

Q: 6. Let (X, T) be a topological space such that every subset of X is closed, then a. (X,T) is an…

A: Given below the detailed solution

Q: 2. Let (X, 1) be any topological space and (X, Tf) be the finite complement topological space. Show…

A: Let (X, τ1) be any topological space and (X, τf ) be the finite complement topological space.Show…

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: Let X be a discrete spaces then * X is never homeomorphic to R O X is homeomorphic to R if and only…

A: Fourth option is correct.