Let x be an infinite set equipped with the indiscrete topology T then (X,T) is compact. True or false

Q: Example. The standard topology Tstd on R" is defined as follows: a subset U of R" belongs to Ttd if…

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Q: Let X be any infinite set with the co-finite topology. Identify which of the following separation…

A: Let X be any infinite set with co-finite topology. Definition:  Let X be any infinite set. The…

Q: Let O be the collection of intervals Ia = (a, ∞) where a R along with I = 0 and I-∞ = R. Does this…

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Q: proof R with the finite-complement topology (F CT) is not continous to R with the usual topology.

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

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Q: A property Is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Given that, A property is said to be a topological property if it is preserved by homeomorphism:…

Q: Let T be the finite closed topology on a finite set X, then * O Tis the Euclidean topology O Tis the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let R be equipped with the Euclidean topology T and let Y =]10,20[. We denote by Ty the induced…

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Q: Let A be any set with the discrete topology. Show that A is normal.

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Q: Show that whether T is a topology on R or not? 2) Let f be a function from a set X in to a…

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Q: Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that…

A: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

Q: Use the Heine-Borel theorem to give an example of a set not compact in R with the usual topology…

A: An example of a set not compact in R with the usual topology and a non-compact set R2 with the usual…

Q: Use the definition of compactness to prove that a closed subset Y of a compact set X ⊆ R is compact

A: In this question, we use the definition of compactness to prove that a closed subset Y of a compact…

Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

Q: The collection of all closed intervals forms.( a subbase for a topology on R. This :topology is…

A: The collection of all closed intervalsfrom a subbase fo a toplolgy on RThis Topology is :-consider…

Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify tha the quotient…

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Q: Let X be an uncountable set and let oo EX be fixed. Let TX = {U E P(X); 0$U or X- U is finite} Let Y…

A: Given: Let us consider the given X to be unaccountable and let ∞∈X be fixed.

Q: Prove that T is the discrete topology for X iff every point in X is an open set

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Second option is correct

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the…

A: Two homeomorphic spaces share the same Topological property If one of them is compact, then the…

Q: Let (X, T) topological space, where X = {a,b,c,d} and T = {X. Ø, {a}, {a,b}, {a,b,c}} Find a) Basic…

A: Given that X, T topological space, where X=a, b, c, d and T=X, ϕ, a, a, b, a, b, c.

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: First option is correct

Q: (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the…

A: Given : X be a set with atleast two points. To prove : Discrete space and indiscrete space ie. (X,τ)…

Q: Which one of the following statements is true? * R with the Euclidean topology and R with the…

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Q: Let T be a topology on X. Assume that T is Hausdorff and let x € X. (1) Show that {x} (and hence…

A: let T be a topology on x. T is a Hausdorff space

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1, 1] with the usual…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

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Q: Let (X,T) be a topological space such that every subset of X is closed, then a. (X,T) is an…

A: (X,T) is a topological space such that every subset of X is closed

Q: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then * O…

A: We are given that X is an infinite set with the finite closed topology T={S subset of X; X-S is…

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: A property is said to be topological property if it is preserved by homeomorphism.

Q: The collection of all left rays and right rays forms a subbase for a topology on R. This :topology…

A: Given , The collection of all left rays &right rays forms  a subbase for a topology on R.

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

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Q: Let R be equipped with the Euclidean topology T and let Y =]10,20[. We denote by Ty the induced…

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Q: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

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

Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is in T, then T is…

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Q: 5) Let 7 be the euclidean topology on R and 7' be the finite-closed topology on R. Define f : (R,…

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Q: Let X be a topological space with the property that, for every topological space Y, every function f…

A: The space X with the discrete topology means every singleton of space is open.

Q: Let X be an infinite set with the countable closed topology T={S subset of X; X-S is countable}.…

A: Connected means it cannot be written as union of disjoint open sets

Q: ii) True or False: The space R is homeomorphic to the closed interval [-1,1] with the usual…

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Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: Given: A property is said to be a topological property if it is preserved by homeomorphism. Suppose…

Q: Let R have its usual topology and let X = {a, b, c}. Define f : R → X by x 0 а, f(x) = с, Is X with…

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Q: Let X be an infinite set with the countable closed topology T={S subset of X :X_S is countable}.…

A: We can show that X is countable. And by taking singleton we can solve the problem

Q: A property is said to be a topological property if it is preserved by homeomorphism. Suppose that R…

A: First option is correct.

Q: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

A: let T be the finite closed topology on any set X.Every subset of (X,T) is connected true or false?

Q: Let X be an infinite set with the countable closed topology T={S subset of X:X-S is countable}. Then…

A: Given that X is  an infinite set with the countable closed topology T =(S subset of X: X-S is…

Q: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then* OX…

A: Gievn that Let T be the finite closed topology and be also the discrete topology on a non empty set…