Let p:Y ->X be a covering map. Prove that any fiber is homeomorphic to a set with a discrete topology.

Q: Let X = {a, b, c, d} with the following topology: ?= {X,0, {a},{b}, {a,b}} Let Y={x,y,z} Consider…

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Q: Let (X, d) be a metric space with X being an infinite countable set. Show that X is not connected.

A: A set is said to be infinite countable if its elements can be counted like natural number but you…

Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Then X" is homeomorphic…

A: Proof-) The Cantor set itself is a cantor space. But the canonical example of a cantor space is the…

Q: 1. Let X = {a,b,c} and B={{a,c}.{b,c}}cP(X). Show that B cannot be a base for any topology r on X.

A: According to the answering guidelines, we shall solve first question only. If you want others to be…

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

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Q: Let E be a subset of X, where X is a metric space. Show the limit points of E are the same as the…

A: In mathematics, the closure of a subset S of points in a topological space consists of all points in…

Q: A ° = A – JA

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Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

Q: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

A: Let (X, d) be a metric space with X being a countable set. Show that X is not connected.

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 X and U be Banach spaces, let C: X→ U be a compact linear map, and let xnx in X. Show that Can →…

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Q: A subset I of a metric space R with the usual metric is compact if and if only it is an interval…

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

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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: Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

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Q: Prove that an affine map is continuous if and only if its linear part is continuous

A: If T:A→Rk is the affine map, where A is the affine span of an affine independent subset of Rn. Let…

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: The set [0, 1] with the discrete metric is compact. True False

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Q: If S is a closed bounded subset of a metric space X, then S is compact.

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Q: Let X and Y be two topological spaces. X is connected and Y is disconnected are they homotopy…

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Q: Find all limit points of a set A = {(2+,2 – |n e N} on the Euclidean plane with the isual metric. Is…

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Q: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

A: Let X={a,b,c} be a set. To show that T={X,ϕ,{b},{a,b}} be a topological space on the set X.

Q: Show that in R with the euclidean metric, any subset of R \Q with more than one element is not…

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Q: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

A: It is given that Xα:α∈J is an indexed family of connected topological space that is each component…

Q: Consider the set A=(x:x>a, a ER)U(x:xsb, bER). Find the topology on R which has A as subbases.

A: given a set  A=x:x>a,a∈R∪x:x≤b,b∈R to find the topology on R which has A  as subbases

Q: Consider the set X = {a, b, c, d} with the discrete topology, construct a basis of space that has no…

A: Consider the set X=a,b,c,d with discrete topology. Consider the basis of this set as follows.…

Q: Let X = R. Define an equivalence relation ~ on X by x ~ y if %3D and only if x – y is an integer.…

A: Let X, ?X be a topological space, and let ~ be an equivalence relation on X. The quotient set,…

Q: Prove the following sets are not compact by finding an open cover that does not have a finite…

A: a) Given-  A set A=−1,1. Explanation- Consider a set Xn=-2,1-1n for n=1,2.... Then clearly,…

Q: Given a metric space M with metric d, verify that any ε-ball is an open set.

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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: Show that any infinite subset A of a discrete topological space X is not compact.

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Q: Let (X, d) be a complete metric space and let F ⊆ X be a closed set. Show that F is complete.

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Q: If x is a limit point of a subset A of a metric space X, then sh

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Q: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

Q: Let x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit…

A: Interior Point: Let A be a subset of metric space X, d. A point x∈A is called an interior point of A…

Q: Consider the spaceZ+with the finite complement topology. Is thisspace a Hausdorff space? Is this…

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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 A CX with the discrete topology. Prove that ô(A) = Ø.

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Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

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Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+…

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Q: Show that it A is subset of X any with the discre te metric then %3D

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Q: Show that the dictionary order topology on the set R × R is the same as the product topology ℝ_d × ℝ…

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Q: f is said to be an open mapping iff the image by f of any open set is open. Let TI be the lower…

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Q: Let X = {a, b, c, d},S = {{a, b, c}, {a}, {b, c, d}}, and T be the topology on X generated by S.…

A: We  use the definition.

Q: Let (X,τ) is a topological space and A ⊆ X. If all subsets of A are closed in X, then set A cannot…

A: Given that X,τ is a topological space. Let's prove this theorem by contradiction. So  X,τ is a…

Q: Let T be the Euclidean topology. Let A = [0,2] and B = [0,1[, then B is clopen in (A,T) where T is…

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

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: Given a metric space M with metric d, verify that any E-ball is an open set.

A: Given that M is a metric space with metric d. Let Ba,ε be an ε-ball with center at a. That is:…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Show f is…

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