(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 or not. dofinition

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: Q1:A) Let X = R and r be the usual topology on R.If Y-[4,6). 1) Determine the open ,closed ,open and…

A: Given that, X=Rand τ is usual topology on R Given that Y=[4,6). 1) 4,5 Is open in the given topology…

Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

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

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: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: Prove that B = (0,1) is NOT a compact set.

A: To prove that (0,1) is not a compact set.

Q: Find two examples for equivalet norms on three dimensional real space R³. .

A: To take two vectors whose norms are the same.

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

A: Sol

Q: 3) Let R be the set of all real numbers with the euclidean topology and f: R → R be continuous…

A: Yes, the image of a continuous function from a path-connected space is always path-connected.…

Q: Let T={X, {a}, {a,b).{a,c,d}, {a,b,c,d}, {a,b,c}} be a topology on X=(a,b,c,d,e). (ii) Find N.…

A: Given : (X, T) be a topological space,  where  X = {a, b, c, d, e} and T = {X, ϕ, {a}, {a, b}, {a,…

Q: Let X and Y be two topological spaces. X is connected and Y is disconnected are they homotopy…

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Q: possible

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Q: Find a topology in which the point x = 5 is an open set.

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Q: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

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Q: Let R be set of real numbers equipped with the standard topology. Clarify with full explanation…

A: Heine- Borel:- A subset of ℝ is compact if and only if it is closed and bounded. Sequential…

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…

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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: (iv) Prove that for any sets A, B,C we have A U (B nc) = (AU B) n (AUC)

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: Define what it means for a topological space to be completely regular,

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

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

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Q: Let x be an infinite set equipped with the indiscrete topology T then (X,T) is compact. True or…

A: An infinite set equipped with discrete topology is not compact.

Q: ACX A + Ø and A + X. Let B = AU fr(A). .show that for the topology T = {X,Ø, A, B} it happens that…

A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology.  Here B=A∪frA To prove for…

Q: Let R, S, and T be sets. Let f: R -» S, and g: S -» T be maps. Assume we know that qf is 1-1 Must f…

A: Given:

Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: (2.2) Let C = {a, b, c, d} and T = {0, {a, b, c} , {b, c} , {a},C} be the topology defined on C. %3D…

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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.