Let X and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the same cardinality

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: 5. If G1,G2 be two given closed sets in a metric space (X, d) prove that: G1 U G2 is closed.

A: We have to prove given result:

Q: Show that R has the same cardinality as the interval (–2, 2) by building a bijection between the two…

A: This is a problem of Real Analysis and Topology.

Q: Thoorem: is not closed set

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Q: We say that a space (X, τ) is not connected if there exist two nonempty openings A, B such that A ∩…

A: Open Set: A X, τ be a topological space. Then the sets in τ are called the open sets of the…

Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

A:

Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

Q: be a closed subset of space Xa

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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: Determine whether is a linearly independent set of vec- 3 11 tors in R3.

A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…

Q: the sets  [0,1] and  [0,1/2)∪(1/2,1] have the same cardinality, give an explicit bijection to show…

A: We will define a bijection from the set J  = [ 0 , 12 ) ∪(12 , 1]   to the set I =0 ,1   First we…

Q: Prove that ifS is a bounded set, then inf(S) = -sup(-S) %3D

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

Q: The continuous image of a connected set is connected. That is, if X is connected and f: X→ Y is…

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Q: 2. Is there a set A that satisfies A = {A}? If yes, exhibit one %3D such. If not, Why not exactly?

A: No, A≠{A} for any set A. Let  A=1,2,3 then number of elements in A are 3. Now,  A=1,2,3 here the…

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

A:

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: Suppose metric space X is not path-wise connected then X is not connected. O True O False

A: Question 9 Answer: as a path connected space of employees connected but converse is not true. So I…

Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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

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Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: If M and N are two norms on space X then prove that M and N are equivalent if and only if a subset…

A: Any two norms M and N on a norm space is called equivalent if  M≤N≤M. in the given question we have…

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

A: A closed convex set in ℝn has a finite number of extreme points if and only if it is the…

Q: Let (X, CA) be a measurable space, f: X (-00, ws) and 1: X (-∞, ∞0) are measurable. Prove that the…

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Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

A: We will answer the first question since we answer only one question at a time. Please resubmit the…

Q: Prove that the unoin of a finite number of bounded sets is bounded.

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

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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: 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: 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 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: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: Show that any subset of bounded set is bounded

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

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

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Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

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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.

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Q: Show that the interval [0,1] is a closed set in R

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Q: Finite topological spaces are always complete. True False

A: Finite topological spaces are always complete. So it is true statement.

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

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Q: Do the sets [0,1] and [0,1/2)∪(1/2,1] have the same cardinality? Either give an explicit bijection…

A: Yes, the sets 0,1 and 0,12∪12,1 have the same cardinality.

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: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

A:

Q: Any two closed intervals has the different cardinality

A: we know that any two closed intervals has the same cardinality

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: Suppose metric space X is not path-wise connected then X is not connected. True False

A: A space is said to be Path connected if we can fine a continuous function between any two points of…