(a) If {Ta} is a family of topologies on X, show that N Ta is a topology on X. Is UTa a topology on X?

Q: Let τ and τ' be two topologies of a set X. Is the family τ∪τ' formed by the open elements of both…

A: Union of two topologies is not necessary a topology.

Q: Is R, equipped with the finite-closed topology, Hausdorff? Provide a proof supporting your answer.

A: Given: ℝ is equipped with the finite-closed topology. Let, ℝ with this topology be Hausdorff and x,…

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: Show that R has the same cardinality as the interval (–2, 2) by building a bijection between the two…

A: This is a problem of Real Analysis and Topology.

Q: i. PROVE THAT THE EXCLUDED POINT TOPOLOGY ON ANY SET WITH AT LEAST 2 POINTS IS NOT HAUSDORFF. ii.…

A: Definition: A topological space X, τ is said to be Hausdorff if for every x, y∈X there exits U, V∈τ…

Q: Let T = {0, En}, where En = {n +1, n + 2, n + 3, ...} with n E N, be a topology on N. Then the set…

A: Ans : False , because

Q: Show that the two intervals (0, 1) and (0, 2) have the same cardinality.

A: In mathematics, cardinality is a measure of the "size" of a set, which can be defined as the number…

Q: let x be an infinite set and let ta be a topology on x in which all infinite subsets of x are open…

A: Given that X be an infinite set and let τ be a topology on X in which all infinite subsets of X are…

Q: Show that compact sets are closed under arbitrary intersections and finite unions. (Here we're…

A: To show that compact sets are closed under arbitrary intersections and finite unions

Q: (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…

A:

Q: Give examples of three relationships that show all possible combinations of minimum and maximum…

A: We need to give examples of three relationships that show all possible combinations of minimum and…

Q: Show

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

Q: 2) Show that the digital line topology on Z is not Hausdorff.

A: Given that Digital line topology on ℤ, we have to prove that it not Hausdorff Therefore, Let a space…

Q: 3) Consider the subsets [0, 1] and [3,4] of R, the set of all real numbers with the euclidean…

A:

Q: Show this collection of subsets a partition of A UBUC {A – (B U C), B – (A U C), C – (A U B), ANBN…

A:

Q: Show that there is no topology on X={a,b,c,d,e} based on the family B={{a,b},{a,b,d},{b,d,e}}

A:

Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

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

Q: 5. If {T,} is a family of topologies on X, show that Ta is a topology on X. Is UT, a topology on X?…

A: First we will prove the result that arbitrary intersection of a topologies is a topology by showing…

Q: Q₂/(a) Let X=(1, 4, 5,7, 8, 9) and o= {(1,4),(4,7, 8}}, show that whether o is: (1) a subbase for a…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: По The finite closed interval [a, b] is compact and no in both the Standard cofinito topology.

A: Soln: The interval [a,b] being closed and bounded in R, thus [a,b] is compact in the standard…

Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: Show that the only Hausdorff topology on a finite set is the discrete topol- ogy.

A: The question asks us to prove that the only possible Hausdorff topology on a finite set is the…

Q: prove that i } X is a Coun table set and P: X>Y is asurjecbive then Y is Countable

A: Note: X denotes cardinality of X, that is number of elements in the set X. If X is countable means…

Q: Let A = {Ba: a € A} be a family of sets and let G be a nonempty set. Prove Gn an (UB.) - U(GOB₂) =…

A: Given: A=Bα : α∈∆ be a family of sets and let G be a nonempty set. To prove: G∩∪α∈∆Bα=∪α∈∆G∩Bα

Q: Give an example of a closed subset S of R^2 such that conv S is not closed

A: An example of a closed subset S of R2 such that conv S is not closed is given below.

Q: 3 let X= {ab,c},B =} {n,b}, {9c3}.5how that Whether B is a %3D %3D base for any topology on X or not

A:

Q: Provide an example to show that connectedness is not a hereditary property.

A: A property P1 is said to be hereditary: If any topological space X inherits the property P1 and all…

Q: Let E be a nonempty subset of an ordered set; suppose a is a lower bound of E and B is an upper…

A: Let X be a non empty set. Let a∈X. Then a is said to be a lower bound of the set X if a≤x for all…

Q: Exercise 10: Show that if A and B are bounded subsets • bounded set. Show that then AUB is a of IR,…

A:

Q: Show that a lattice is distributed if and only if for any elements a, b, c in the lattice (avb)^с ≤…

A: To Show- Show that a lattice is distributed if and only if for any elements a, b, c in the lattice a…

Q: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with the euclidean…

A: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with theeuclidean topology.…

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: Question 3. Consider the usual topology U on the set of real numbers. Describe the relative topology…

A:

Q: shows that there is no topology in X based on the family B = {{a, b}. {a, b, d}, {b, d, c}}

A:

Q: If G(sub alpha) is a collection of closed sets, then the intersections of the G(sub alphas) is a…

A:

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: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

A:

Q: Show that is not compact for the Standard topology but is compact for the cofinite topology.

A:

Q: Ris not connected if T is the trivial (usual) topology O the lower limit topology the indiscrete…

A: We know that

Q: Prove box topology and product topology are equal in the finite cas

A:

Q: How can I show that the collection  A composed of empty set and all cofinite subsets of X Is a  sets…

A: