Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is homeomorphic to Rif and only if X is infinite O X is never homeomorphic to R X is homeomorphic to R if and only if X is çountable

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: 1. Show that if f is a bijection from a set X onto a set Y then f is a bijection from Y onto X.

A: Please post the multiple questions separately. Here I answered question (1) only as per our policy.

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

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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: Let Y be an infinite subset of a compact set X c R. Prove Y' # Ø.

A: Y' denotes the derived set of Y i.e. the set of all limit points of Y. We just need to show that any…

Q: Let X be a topological space having an open cover U1,..., Un with the property that all k-fold…

A: Solution

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: Which of the following is true: Every bounded subset of (R, Tusuat) is compact. Every closed subset…

A: We have to find correct statements out of given statements.

Q: Let X be a topological space. Show that the following are equivalent. (i) X is disconnected. (ii) 3…

A: Let X be a topological space then, To prove: Following statements are equivalent: (i) X is…

Q: 3- In a complete metric space (X, d), If A C X such that, A = A then A is complete as a sub space of…

A:

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: Provethat amappi'ng from a metric SpaceXto metri'c space X is con tinuok ifand only f the inverse…

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Q: Let (X.d) be mnrtric space and X is closed then X is O complete Ono decsion Onot complete

A: A metric space X is said to be complete if every cauchy sequence in X is convergent. That is, every…

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: 5. State whether each of the following statements is true or false, in either case substan- tiate/…

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Q: Prove if {Galpha}alpha e A is a collection of closed sets in a metric space, then…

A: Recall that arbitrary union of open sets in X is open in X.

Q: A normed space X overR is separable if it has a conutable dense subset 0 صواب ihi

A: The given statement that " A normed space X over R is seperable if it has a countable dense subset "…

Q: Let (X,7) be a topological space and ACX. Show that (n) A is open if and only if Int(A) = A. (b) A…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

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

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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…

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Q: 3. Prove that if G is an open set dense in the metric space (S, d) then SG is nowhere dense in S.

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Q: In an indiscrete space X * O the only dense subset of X is X O every subset of X is dense in X a…

A: Suppose X is a set then a topology τ defined on X is said to be indiscrete if τ={∅, X}

Q: Let X and Y be two locally compact spaces, then X*Y is locally compact. (Recall that: X is said to…

A: This is a problem of topology.

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

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Q: Let X be a discrete spaces then O X is homeomorphic to R if and. only if X is countable O X is…

A: I have provided a solution in step2

Q: A and B are open subsets of X with Cl(A) = X = Cl(B).

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Q: Prove that the set {x e R : 10/x - x > 0} is bounded. Prove that the set {x E R : x² – 25x > 0} is…

A: To prove that the set x∈R:10x−x>0 is bounded. To prove that the set x∈R:x2−25x>0 is unbounded.

Q: let C be a compact subset Tn a Haus dorft topolopfcal space (K,R) and bex a If bac then prove that…

A:

Q: Prove that G is open in a space X if and only if Gnà = GNA for every subset A of X. topology problem

A: Let G be open in a space X and A be any arbitrary subset of X To prove G∩A¯¯=G∩A¯ For any subset A…

Q: Let Q be the set of all rational .numbers a) If T= {0}U{ACQ: Q-A isfinite} prove that (Q, 7) is…

A: For the solution follow the next steps.

Q: Show that for any space A there does not exist an injection P(A) → A.

A: To prove that for any space A, there doesn't exist any injection PA→A Let us consider the contrary.…

Q: Let X be an infinite set with the countable closed topology T-{S subset of X; X-S is countable}.…

A: Given that X is an infinite set with the countable closed topology T-S subset of X; X-S is…

Q: 2. Prove that if K is compact in R and F is a closed subset of K, then F is compact using (a) the…

A: To Prove- Prove that if K is compact in R and F is a closed subset of K, then F is compact using…

Q: Every finite space is: (a) Compact space (b) Not compact space (c) Hausdorff space. (d) Non of above

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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

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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: Let (X, d) be a metric space. a) Assume that A is a (finite or infinite) collection of open sets.…

A: To orove the required topological properties of open sets in a metric space (X,d)

Q: 1.Any open interval in R is an Open set in R 2.The {0} is open in R with discrete metric

A: We will use the fact of open ball

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

A:

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

A: Given that, X is a discrete space. To be homeomorphic X has to be of same cardinality as ℝ and so X…

Q: A space X is locally connected if and only if for every open set U of X, each comj

A: Suppose that X is locally connected. Let U be an open set in X and let C be a component of U. If x…

Q: Prove the following: Suppose that {X; : i e I} and {Y; : i e I} are indexed families of topological…

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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: Let K and L be two compact subsets of a metric space, then: O KUL and KNL are both compact. O KUL is…

A: (i) Any open cover of X1 ∪ X2 is an open cover for X1 and for X2. Therefore there is a finite…

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.

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.