(2) Let (X,r)be a topological space, If for every A open set, and x E A", 3 Ux and Vy open sets, such that x E Ux, AC Vy, and Un V, = 4. The space is called: (a) Hausdorff Space (b) Regular space (c) T1-space (d) Normal space

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

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A:

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

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A: Assume E is disconnected. Then there exist two open set U and V in E such that 1. U∩V=∅ 2. E=U∪V 3.…

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Q: 5. Let (X, d) be a metric space. i) Show that the open ball B, (ro) = {x € X : d (T, xo) < r} is an…

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

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A: 2. Construct two metric spaces  and  which are homeomorphic, but  is complete and  is not complete.

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A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

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A: Use definition of disconnectedness mentioned below.

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A: d. neither connected nor pathwise connected space.

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A: Given: X,d is a metric space, x∈X and A⊂ X is a non-empty set. To Prove: dx,A=0 if and only if every…

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Q: ctors - Z. S. Madhi

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Q: Q. Let S = {(x, y), x² + 4y2 = 4} CR2, then S - {point} is homeomorphic to – y? = 4} a. {(x, y), 4x2…

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

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A: As there are two proofs asked, I will solve only the first one. Let M1 and M2 be connected spaces.…

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A: We prove the result by mathematical induction We know the triangle inequality of metric space If x,…

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

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

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A: Given the set X contains more than one element.

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Q: ANB +P

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A: Given : ℤ , τ is indiscrete topological space. To show : To show whether ℤ , τ is regular , T0, T2,…

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Q: disprove the following AUB s connected set then either A set prove on or B is conniedted

A: Since it is a multiple questions. As per our company guidelines, we have to solve the first one. For…

Q: Amongst the various possible products of four topological spaces A, B, C and D, show that (A × (B ×…

A: We shall solve this in next step