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

Q: Show that every metric space is an Hausdorff space.

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Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

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: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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Q: Elaborate on the role of duality in topological spaces.

A: Introduction:Topology is a branch of mathematics that studies the properties of spaces and the…

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

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

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

Q: Find an example of a topological space that is seperable and first countable but not second…

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Q: 1 -1 -1 1 -1] Let A -1 -1 1 1 1 Find a non-zero vector x in the null space of A. -1 -1 -1 -1 X =

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Q: what is a proper space

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Q: Explain with an example of a topological space which is singleton that is open but not close and…

A: Example in refer of not  open but closed .Take , Here A is singletone set which is closed but not…

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

A: This is a problem of Functional Analysis.

Q: Give an example of a norm space.

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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: Define what it means for a topological space to be completely regular,

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Q: Prove that in any metric space, an ɛ- eighborhood of a point is an open set.

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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: 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: Tet fiR*-o R. be dafined by it ² tyz if X=y =O Show that f is qutinyous on IR

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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…

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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