Suppose metric space X is not path-wise connected then X is not connected. O True O False

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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Q: Show in the discrete metric space: 1. All sets are open (especially single points);

A: Let X be a non-empty set and d be discrete metric on X

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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Q: We say that a space (X, τ) is not connected if there exist two nonempty openings A, B such that A ∩…

A: Open Set: A X, τ be a topological space. Then the sets in τ are called the open sets of the…

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: 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: Every closed and bounded subset of X is compa The subset {z EX: |||| ≤ 1} of X is compact... X is…

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Q: Let X={a, b, c, d, e} and let T={X, non zero, {a}, {a, d}, {a, e}, then (X, T) is neither Hausdorff…

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Q: 2. Determine whether the following multigraphs are Eulerian, semi-Eulerian or neither. Explain your…

A: Explanation we have to shown whether the given multigraphs are Eulerian , semi-Eulerian or neither…

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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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: show that (0.1)with usual topology is connected

A: Sol

Q: Prove that any ball (open, closed, half open) in R³ is path connected. or or

A: Please check the answer in next step

Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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Q: Show that if A is closed in X and B is closed in Y then A × B is closed in X × Y

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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: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: QUESTION 9 Suppose metric space X is not path-wise connected then X is not connected. True False

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Q: Hausdorff space.

A: Given: A set X with co-finite topology and  Hausdorff space. To Find: Under what condition a set X…

Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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Q: Let lx, T) be a topological Space and A, B S X- if A is open, then show that An CIIB) s CLLANB). if…

A: Given X, τ be a topological space and A, B⊆X. Suppose, A is open then An arbitrary x∈A∩clB implies…

Q: Provide an example of a topological space that is connected but not path-connected,why?

A: We use definition of connected and path connected sets. We use contradiction method.

Q: X is not open, must it be closed? Either prove that every subset E that is not open must be closed,…

A: We will show it by giving example

Q: if

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

Q: The usuall metric space (Rd)isComplete ***

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Q: Let X = {a, b, c, d,e} and let T = {X, Ø, {a}, {a,d}, {a,e}}, then * (X, T) is not Hausdorff but it…

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: Prove by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

A: In this question, we have to show that open balls in ℝn with the usual metric are not compact by the…

Q: Suppose U and v are conformally equivalent. Prove that if U is simply connected, then so is V.…

A: Given that U and V are conformally equivalent that is there exist a mapping f from U to V such that…

Q: 7 The set M={xEN:x<÷} with the usual metric is connected.

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Q: (3) Let X be a connected, locally path connecte

A: Let X be a connected, locally path connected space and the functionπ1(X) is finite . We have to show…

Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: Show that all open balls are bounded subsets of a metric space.

A: Solution: Let M=X,d be a metric space. For an arbitrary point a∈X, the open ball Bεa where ε&gt;0 is…

Q: 8. Prove that a matroid is uniform if and only if every circuit meets every cocircuit.

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Q: show by an example that a nonempty proper subset of S in the metric space (S, d) can be both open…

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Q: (a) A discrete space with more than one point is disconnected. (b) Any trivial space is connected…

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Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

Q: For any metric space (X, d) and x EX, the singleton set (x) is closed. Select one: OTrue O False

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Q: Give all boundary points of D.  Is D open, closed or bounded? Justify all your answers.

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Q: Consider X = {a1, a2, ..., an} Can a topology of X with 3 open and one with 4 open be equivalent?

A: We have to verify the given statement.

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: a nonempty compact set of IR. sup Ke K. be

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Q: 1. Prove that any ball (open or closed half open) in R³ is path connected. or

A: Definition: A topological space X is said to be a path connected space if given x, y∈X there exists…

Q: Prove that any ball (open or closed or half open) in R³ is path connected.

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Q: If e-ispure imaginary W hat Restriction is placed on z?

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Q: Suppose metric space X is not path-wise connected then X is not connected. True False

A: A space is said to be Path connected if we can fine a continuous function between any two points of…