Define metric space and define its four properties

Q: Show that every second countable (A2-space) space is separable.

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Q: Exercise. Is it possible for a metric space (X, p) to contain more than one point and be such that…

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Q: Exercise. Show that in R with the euclidean metric, any subset of R\Q with more than one element is…

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Q: Is the euclidean space R³ is a complete metric space

A: We have to check whether euclidean space R³ is a complete metric space or not .

Q: 3. Let (X, d) be a metric space and x1, x2, ... , Xn, ... a sequenc

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Q: are closed.

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Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: 6. Show that a compact metric space X is locally compact.

A: The given problem is related with metric space. We have to show that a compact metric space X is…

Q: Compute the dimension of Span (v1, V2, V3, V4, V5, V6 ).

A: Follow the procedure given below.

Q: Let be a compact Hausdorff space, then find the necessary and sufficient condition for space to be…

A: Given that a compact Hausdorff space  To Find : Necessary and Sufficient condition for space to be…

Q: Let X be a topological space

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Q: Every metric space is a hausdorff space .

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Q: What is norm space

A: In mathematics, a normed space or a normed vector space is a vector space over the real or complex…

Q: Show that the real line is a metric space.

A: Answer is mentioned below

Q: A Hilbert space is * O Inner or complete O Only complete O Only normed O Metric and complete O…

A: Define Hilbert space and select appropriate option.

Q: Define the term dimension of a space.

A: In mathematics, the dimension of space means the minimum number of coordinated to specify or located…

Q: Prove the following: The countable product of metrizable spaces is a metrizable space.

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Q: Show that finite dimenshional inner product space is hilbert space

A: An n-dimentional inner product space is not Just Topologically isomorphic to Rn. It is isometric to…

Q: Verify that the discrete metric is actually a metric.

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Q: Exercise. Every point in a discrete metric space forms an open ball.

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Q: Please define compact space and the clousure of .a set

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Q: Let two metrics be bi-lipschitz, prove it is an equivalence relation

A: Find the attachment for the solution.

Q: topological space a

A: Presumably, the closure of Q in R under the usual topology. It suffices to show that for every real…

Q: Any cover of a metric space is open.

A: Any cover of a metric space is open_____?_____

Q: Every inner product space is metric space and normed space O metric space but may be not normed O…

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Q: (a) Prove that the metric space (2,11) is complete,

A: Complete Metric Spaces:- A Metric Space (X,d) is said to be complete iff every Cauchy sequence in X…

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

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Q: Exercise. Prove that every finite subset of a metric space is closed set.

A: We have to Prove that every finite subset of a metric space is closed set.

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: A finite set in a metric space cannot have accumulation points.

A: We have to prove that A finite set in a metric space can not have accumulation (Limit) point.

Q: What is Euclidean Space??

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Q: every finite subset of metric space is compact. ture or false?

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Q: Determine the dimension of the space spanned by the set

A: Given,               S=12-13, 1002, 28-48, 1111, 3306

Q: ": Prove that ( R", ||||) is complete metric space.

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Q: What is Sierpinski Space? Also give example

A: Sierpinski Space is  a topological space. Now to define a topological space we have to define a…

Q: Every closed subset of a connected metric space is connected. O True O False

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Q: The set {0,3} with the usual metric is connected as union of connected sets {0} and {3} True False

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Q: (a) Show that every metric space is T2-space.

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Q: Q7: Prove that ( IR", II-||) is complete metric space.

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Q: If X is a metric space and Y C X, then Y is a metric space.

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Q: show that a set of natural numbers N ,with Euclidean metric is Separable

A: We have to prove that a set of natural numbers N, with Euclidean metric is Separable.

Q: Show that a union of any finite points of bounded subsets of a metric space is bounded

A: One way to prove this result is to take each set in the union and pick a single point to be its…

Q: Find two examples for equivalet norms on three dimensional real space

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Q: Theorem 9.8. A metric space is Hausdorff, regular, and normal.

A: The objective is to show that every metric is hausdorff, regular and normal. Let x≠y be points of a…

Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…