Ametric space (x,d) is complete if and only ifx has the cantor intersection property

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

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Q: If a CW-complex X has one 0-cell, two 1- cell, one 2-cell, and no other cells, then the space X is…

A: We need to determine the below statement is true or false. Statement is : If a CW-complex X has one…

Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

Q: For the given matrix A What is the rref of A. What is the rref of AT A = [1 2 1 1 -1 21 0302 5 1 4 5…

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Q: In cartesian product, prove that (A-B)XC=(AXC)-(BXC) A,B,C are sets

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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: Let C denote the set of all ordered pairs (a, b) with a,b & R. L.e., C:= {(a,b): a,b € R}. Define…

A: NOTE: According to guideline answer of first three related subpart can be given, for other please…

Q: Prove that B = (0,1) is NOT a compact set.

A: To prove that (0,1) is not a compact set.

Q: Find two examples for equivalet norms on three dimensional real space R³. .

A: To take two vectors whose norms are the same.

Q: al let (xt) betopology space and BCY then B is closed in y iff there is closed set CinX st B-COY

A: Let (X,Γ) be topology space and B⊂Y To Prove: B is closed in Y iff there is a closed set C in X s.t…

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

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Q: Let X And Y be two distrecte spaces. Then X is homeomorphic to Y if and only if X and Y are both…

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Q: 4={(-21.r):reR} II) B=(2r.-31): te R} III) C={(2r+3,-41) re R} I)

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Q: Show that if a Banach space X is non-trivial, X + {0}, 3). then X' # {0}.

A: According to the given information, it is required to show that:

Q: Define the Cartesian product of sets X and Y? How is this Cartesian product denoted?

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Q: Q1) Prove or Disprove: Let L(F) be a linear space and A,B are two convex subsets of L. Then, AXB is…

A: The problem is related with convex set. Given that,                 LF  is a linear space  and A, B…

Q: Can you show an example of Null space?

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Q: Let X = {1, 2, 3, 4, 5} and Y = {7, 11, 13} %D are two sets. find R = {(x, y): x EX and * y EY and…

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Q: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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Q: If A and B are distinct points in 3-space then |AB| = - Select one: True False

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Q: Finite topological spaces are always complete. True False

A: Finite topological spaces are always complete. So it is true statement.

Q: ACX A + Ø and A + X. Let B = AU fr(A). .show that for the topology T = {X,Ø, A, B} it happens that…

A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology.  Here B=A∪frA To prove for…

Q: Show R³ is a banach space.

A: Banach space: A complete normed linear space is called Banach space. Given: To show:  is a Banach…

Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: Show That R is Banach Sface.

A: To Show -

Q: Show that a space X is Hausdorff if and only if the diagonal {(x, x)|x € X} is closed in X2.

A: We use basic topological definition to prove this.

Q: Show that if X and Y Hausdorff spaces, then so is the product space X × Y. are

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Q: Show that the closure of a connected space is connected.

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