nempty set XC Rn is connected if and only if for every disjoint open sets U and V in Rn such that X CUUV, we have X CU or X CV. Exercise 6.61. Let X be a connected subset of Rn. Prove that if ECR", XnE ‡ Ø, and Xn0E = Ø, then X CE°. Theorem 6.62. [a, b] is connected.

Q: Suppose X and Y are subsets of some universal set. De Morgan's laws provide expressions for the…

A: X∪Yc=Xc∩Yc

Q: Exercise 2. Assume that (X, T) is a topological space. a) If A, B are subsets of X with AC B, show…

A:

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A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

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A:

Q: A – Ö B, = Ü(A – B.). Ü(A – B;). i=1

A:

Q: Definition. Let X be a topological space and suppose xo is a point in X. A map exo : [0, 1] → X that…

A: To prove the theorem,  First define the homotopy function  H:I×I→X by Hs,t=α2st+1    0≤s≤t+12y…

Q: Let ā = ( – 5, — 4, 5) and - 4, 5) Find the projection proja b 6 and 7 = (3, 1, 0). =< of b onto a.…

A:

Q: Prove the following parts of Theorem 2.2.1: For all sets A , B and C A ⊆ A ∪ B

A: Assume xϵA

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A:

Q: Ex 4.) Let Sbe a non empty bounded set in 1. aj Let azo, and let a S:= {as: SES}. Prove that inf…

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A: Identities of the sets

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A: Topology Question

Q: 9. Let X be any nonempty set, and let B = {{x}: re X}. Show that B is a base for a topology on X.…

A: Given X be any nonempty set, and B =  x : x∈X

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A:

Q: Let U CR. Prove U is open if and only if U = U°.

A: We have to solve given problem:

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A:

Q: Exercise 3.3.15. Prove or find a counterexample to the following statement. Let A, B and C be sets.…

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