5. If {T,} is a family of topologies on X, show that Ta is a topology on X. Is UT, a topology on X? Discuss your answer.

Q: If  X is a countable set show that every topology on X is second countable.

A: To Show: If  X is a countable set show that every topology on X is second countable.

Q: Suppose S be an open connected subset in arbitrary topological space X. Whether S is path connected…

A: If S is an open connected subset of a topological space X, then it is path connected. A subset S of…

Q: Let T = {0, En}, where En = {n +1, n + 2, n + 3, ...} with n E N, be a topology on N. Then the set…

A: Ans : False , because

Q: 6. Show that the span of this set is all of R?. {}}

A: Let S=v1,v2,...,vn be a set of vectors in a vector space V(F) . The vector space V is spanned by S,…

Q: 4. Let A and B be two closed subsets of the connected topological space X. Suppose that X = AUB And…

A: Method of contradiction : suppose A is disconnected ,then there are two disjoint closed non empty…

Q: 3. Determine if the matrik 4 is in the span of the set S = D

A:

Q: 1. Let X be a topological space and be an equivalence relation on the elements of X. Define : X→ X/~…

A: Please check the answer in next step

Q: 2-2. Let X = {1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic…

A:

Q: (1) Let X be a set and tEX, let t={GCX/ t€G or G° finite }, show that whether t is a topology on X…

A:

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

A: Sol

Q: Show that there is no topology on X={a,b,c,d,e} based on the family B={{a,b},{a,b,d},{b,d,e}}

A:

Q: 4 ,3 ,2 ,1 = X آقا -X L find The Possible Topology That Contain sus { 1123 , 2, 3, {\/ 3 ، 4 seas

A: It is given that,  X = {1, 2, 3, 4}. We have to find the possible Topology on X that contain the…

Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: 3 let X= {ab,c},B =} {n,b}, {9c3}.5how that Whether B is a %3D %3D base for any topology on X or not

A:

Q: Is T ={∅, X, {1, a}, {1, b}, {2, c}, {1, a, b}, {1}} a topology on X = {1, 2, a, b, c}? Why?

A:

Q: possible

A:

Q: Show that the lower limit topology on R is strictly finer than the standard topology on R. That is:…

A:

Q: if

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

Q: Question 3. Consider the usual topology U on the set of real numbers. Describe the relative topology…

A:

Q: shows that there is no topology in X based on the family B = {{a, b}. {a, b, d}, {b, d, c}}

A:

Q: Let X = {1, 2, 3, 4, 5} and Y = {7, 11, 13} %D are two sets. find R = {(x, y): x EX and * y EY and…

A:

Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: How can I show that the collection  A composed of empty set and all cofinite subsets of X Is a  sets…

A:

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: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

A: Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X…

Q: 3) let X be discrete topology and Ac X, find A ?

A:

Q: Q.No.2 Let X be a universal set and A, B be the subsets of X.Then prove the following. (a) ((A*…

A:

Q: Let (R, T) be the set of real numbers with the cofinite topology. (See Example 2.2.5.) i tud…

A: Given: I={U⊆ℝ:X-U is finite} (a) Consider a finite set, say ATo Show:A is closedequivalently we show…

Q: Show that X is connected if and only if the only clopen subsets are Ø and X.

A:

Q: If X is any set, the collection of all one-point subsets of X is a basis for the discrete topology…

A: Given : Let (X, T) is a discrete topological space on the set X. To prove : We have to show that all…

Q: If T1 and T2 are two topologies on the same space X, then

A:

Q: Let n e N. Suppose An = {n,n +1, n +2,...} and T = {0, An}nɛN• (a) Show that T is a topology on N,…

A: As per the guideline, we are supposed to do the first three subparts of a problem. Kindly repost the…

Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

Q: Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b.…

A: A topological space is an ordered pair {\displaystyle (X,\tau ),} where {\displaystyle X} is a set…

Q: Suppose X has the discrete topology. Then the infinite product X with the product topology is also…

A: It is given that X has the discrete topology. Check whether the infinite product Xw with product…