5) Let Xbe an arbitrary space and let J, and I be two topologies on 1 2 it. (i) Is T, U T always a topology on X? If so, prove it. If not, give a 1 2 counter-example. (ii) Is T, N I always a topology on X? If so, prove it. If not, give 1 a counter-example.

Q: 3. Prove that the connected components of a space X1,..., X, are sets of the form E x ...x En where…

A: Let Y be the connected components of a space X1X2...Xn. The connected components of X1X2...Xn are…

Q: 11. Let (X, d) be a pseudometric space, and suppose d has the property that d(a, b) >0 whenever ab.…

A: 11) To prove that every finite subset of X is closed, we need to show that its complement is open.…

Q: (a) Is the set S = {1 + x + x2, 1 − x3,  x + x2 + x3,  x3} ⊂ P3 linearly independent? If so, prove…

A:

Q: Every Hausdorff space is hereditarily Hausdorff.

A: We have to prove that every Hausdorff space is hereditarily Hausdorff. Consider a Hausdorff…

Q: Theorem 5.9. Let X be a 2nd countable space. Then X is separable.

A: we have to show that , x be a 2nd countable space then x is separable

Q: 2.2 Base for a Topology 39 Let X be the space of reals with the topology T = {X, [0, 1], 0} (Example…

A: The given topology is T={X,[0,1],∅}. The given sets are A=[-1,1], B=[2,3].   The derived set is the…

Q: Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and…

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.…

Q: 2. Consider the following mappings: • A€ L(R³) such that A(@1, a2, a3) = (a1 + a2 + a3, R CIP3) cuch…

A: (a). 1. Aa1,a2,a3=a1+a2+a3,0,0 Consider b1,b2,b3, Ab1,b2,b3=b1+b2+b3,0,0 1.…

Q: Let A, B and X be topological spaces. If A can not be embedded into B then A × X can not be embedded…

A: Let's first address the claim that S1 cannot be embedded into R. This claim is incorrect. The circle…

Q: Determine the null space of A and verify the Rank-Nullity Theorem. 1 -4 9 9 -7 A = -1 2 -4 1 5 -6 10…

A:

Q: d T = {0, {a}, {a, b}, {c,d}, {a, c, d}, {a,b, c, d}, X}. (1) Is this space To? Why or why not? (2)…

A:

Q: Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and…

A: Use definition of disconnectedness mentioned below.

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: Theorem 2.47. Suppose B is a basis for a topology on X and Yc X. Then By := {BnY || BE B} is a basis…

A: Given that, BY=Y∩BB∈B is a basis for the subspace topology on Y⊂X. Now, we know that, TY=B∩YB1 is…

Q: Exercise 4.6. (1) Consider R² with the standard topology. Let p ER² be a point not in a closed set…

A: part(a) Let A be a closed set in ℝ2. Let a point p∉A⇒p∈Ac As, we know A is a closed set thus Ac is…

Q: (a) State the definition of an antichain A of a set S.

A: As per guidelines only first question is solved. For remaining question kindly repost the question…

Q: If A, B are subsets of the topological space X show that Ext(AUB) = Ext(A)n Ext(B).

A: Given: A and B are subsets of the topological space X. To Prove: ExtA∪B=ExtA∩ExtB

Q: 5. Let X be the space of reals with the topology T = {X, [0, 1], 0) (Example 2.1(b)), and let A =…

A: The given topology is T=X,0,1,∅. The given sets are A=-1,1, B=2,3.   The derived set is the set of…

Q: 5. In any metric space (M, D), prove that D(a1, an) ≤ D(a₁, a2) + D(a2, a3) + + D(an-1, an), for a1,…

A: Given M,D is a metric space.   The triangle inequality holds in a metric space, which means for…

Q: 2. Answer the questions with Yes or No, and then explain why yes or why not. Justify your answer. a)…

A: Answer :

Q: We note that ℤ2 which means ℤ*ℤ. Let d: ℤ2 *ℤ2→ℝ≥0 be the function that is given by…

A: The given in this question is a function d that takes two points (x0,y0) and (x1,y1) in the set of…

Q: (c) let t be indiscrete topology on set of integers Z, show that whether (Z, t) is regular space or…

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

Q: Theorem 2.15. Let (X,T) be a topological space, and let U be an open set and A a closed subset of X.…

A: