If X #0, compare the following topologies on X: a) The co-finite topology 7, and the co-countable topology 7, if X is infinite set. b) The co-finite topology 7, and the co-countable topology 7, if X is a finite set.

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: In the discrete topology on the set X, every set is open and it also follows that every set is…

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Q: If a metric space M is the union of path-connected sets Sa, all of which have the nonempty…

A: Given M is a metric space which is union of path connected sets Sα. All Sα have the non empty path…

Q: show that the Cantor set is a closed subset of R and has measure zero.

A: We know the definition of Cantor set, The Cantor set is defined as the intersection C = ∩n=1∞Cn ,…

Q: Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected…

A: Let's examine the  problem for path-connectedness example of a set in R2 with the usual topology…

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: Use the Heine-Borel Property from (5.1), which states that a set of real numbers is compact if and…

A: See solution below. Here we used definition of bounded and closed sets.

Q: Give two sets that are equal. Prove that they are indeed equal by using the "Axiom of…

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Q: If S is compact and D is in S is a closed set, then D is compact. Use the definition of…

A: Let S is compact and D be a closed subset of S. We are to prove that the set D is compact in S. Now…

Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Suppose a set M is closed and bounded. Prove that M has a maximal and minimal element.

A: Suppose a set M is closed and bounded. Then we have to show that M has a maximal and minimal…

Q: (b) Show that the continuous image of compact set is compact but the convers need not to be true.…

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Q: Show

A: We prove this statement by using definition of Borel Set.

Q: The continuous image of a connected set is connected. That is, if X is connected and f: X→ Y is…

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Q: 2- Prove that H, is convex set.

A: Let us look at an arbitrary closed half-space defined as H+={x∈ℝn:aTx-b≥0}, representing a half…

Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: If A C BỊ U B2 where B1 and B2 are disjoint open sets and A is compact, show that An B1 is compact.…

A: To Show- If A ⊆ B1 ∪ B2 where B1 and B2 are disjoint open sets and A is compact, show that A ∩ B1 is…

Q: Is the Cantor set closed? Is it compact? Explain!

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Q: Suppose that A and B are disjoint finite sets. Prove that AUB is finite.

A: Here are given sets A and B are finite sets. Then A=m, B=n  where m, n any fixed positive integer.…

Q: For any infinite set X, the co-countable topology on X is defined to consist of all U in X so that…

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Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

Q: Continue in this fashion, removing the middle third from all of the segments of set Ak, thus…

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Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: Show that any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

Q: In any metric space, the only open and closed set at any time is the empty set and X only True False…

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Q: Decide whether the following propositions are true or false.If the claim is valid, supply a short…

A: A compact set is always closed and bounded.

Q: Use the definition of compact set to prove that the set [0,1] is compact

A: We know that a set E is compact if every open cover of E has a finite subcover. We suppose, that O…

Q: Suppose A is a set and R is a reflexive and transitive relation on A. Show that the

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Q: Tet X and Y be infinite sets prove that XUY is infinite

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Q: if

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

Q: 5.3.8 Prove that if A CR is a finite set, then A is well-ordered.

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Q: In R with the cofinite topology, every preopen set is open

A: As we know that , In ℝ with discrete topology of every semi open set is open.

Q: Question No 1 (8). Prove that the discrete topology on an uncountable set does not Satisty the…

A: Note- Let X be a topological space, then X satisfies the second axiom of countability or is second…

Q: Let C be a nonempty subset of a bounded set D. Why does inf C and sup C exist? Show that (c) inf D ≤…

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Q: The closare of a set is smallest closed set containing s صواب ihi

A: Let S be a subset of X: S⊆X. Let M be the closure of S. Then M is the smallest closed set which…

Q: A set L cR is convex if:

A: In this question, we have to find the correct option for the convex set L⊂Rn  here we use the direct…

Q: 2. For any n E N, let G, denote the interval (-). Prove that each G, is an open set in R and each G,…

A: Here Gn=-1n,1n     so, G1=-1,1           G2=(-12,12)           ..           Gn=-1n,1n Since all Gn…

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: Construct a family of sets that demonstrates R is not compact

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Q: 4. If X #0, compare the following topologies on X: a) The co-finite topology Tf and the co-countable…

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Q: 1. Prove that a set SCR is everywhere dense if and only if S intersects every non-empty open set in…

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Q: Give an example of a set in R ^ 2 with the usual topology that does not have any subset (not…

A: Given that R2 with the usual topology. The aim is to find an example of the set  R2 with the usual…

Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

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Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

Q: A collection of open sets in R with the standard topology whose intersection is not open.

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Q: Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E)

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Q: Consider the set X={a,b,c,d} construct all the possible topologies of X that have only three open…

A: Given: The set X=a,b,c,d To find:  all the possible topologies of X that has only three open…

Q: On a set X, consider the collection consisting of four of its subsets, given by Γ = {X, ∅, A, B},…

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