Finite topological spaces are always complete. True False

Q: A topological space is Hausdorff iff limits of all nets in it are unique

A: This is a problem of Topology.

Q: topological space X is the intersection of all its closed neighbourhoods. A regular if and only if…

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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: show that any infinite discrete topological space is not compact

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Q: A topological space is Hausdorff iff limits of all nets in it are unique-

A: This is a problem of Topology.

Q: Every totally  bounded metric space is compact

A: This statement is not always true.

Q: a) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

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: Prove that the quotient of a locally connected topological space is locally connected.

A: We have to prove that Let q:X→Y be a quotient map, X is a locally connected space. Show that xY is…

Q: Every Hausdorff space is hereditarily Hausdorff.

A: We have to prove that every Hausdorff space is hereditarily Hausdorff. Consider a Hausdorff…

Q: (b) Show that the continuous image of compact set is compact but the convers need not to be true.…

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Q: Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if…

A: This is a problem of topology.

Q: Define a compact space.

A: The objective is to give the definition of a compact space.

Q: A metric space is sequentially compact if and only if it is totally bounded and complete. prove it

A: We need to prove that any given metric space is sequentially compact if and only if it is totally…

Q: 36) Prove that every product Space is inner metric Space- Discuss its Converse

A: Suppose that V is an inner product space over a field F. To prove that the V  is a metric space. To…

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

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Q: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: Arbitrary unions of closed sets are closed.

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

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Q: Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point.

A: We have given that, X is Compact space.

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: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

Q: Any cover of a metric space is open.

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

Q: If (X.r,) and (X,r,) are T.-spaces, then (X.r,nr,) is also a T,-space. GOOD LUCK

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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: Theorem 2.25. Suppose that (X, p) is a metric space. (a) The intersection of any collection of…

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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: 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: Every finite space is: (a) Compact space (b) Not compact space (c) Hausdorff space. (d) Non of above

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Q: Prove or give counter example an infinite product of discrete spaces may not be discrete

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Q: 1. prove every 2nd countable space is Lindelof space. 2. Every Lindelof topological space induced by…

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Q: Can you show an example of Null space?

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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: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

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

Q: c spaces. Show that X x Y and Y

A: The sup-metric d1 on X×Y is defined as d1x1, y1, x2, y2=max dx1, x2, ey1, y2. Similarly, the…

Q: a discrete space

A: We know that in a discrete topological space X, every point in X is an open set.

Q: Show that topologically completeness is weak hereditary property.

A: In this question, we have to show that completeness is a weak hereditary property.

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

A: We use contradiction method.

Q: The closure of a compact subject of a regular space is compact.

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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: A discrete metric space X is separable if and only if X is countable

A: Given: To prove: A discrete metric space X is separable if and only if X is countable

Q: Every compact, Hausdorff space is normal.

A: To prove: Every compact, Hausdorff space is normal. Let X be a compact Hausdorff space. Let A,B⊂X be…

Q: 2.2 Is any discrete space complete? Give an example to substantiate your answer.

A: Yes, any discrete space is complete.

Q: Let K and L be two compact subsets of a metric space, then: O KUL and KNL are both compact. O KUL is…

A: (i) Any open cover of X1 ∪ X2 is an open cover for X1 and for X2. Therefore there is a finite…

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

A:

Q: Suppose X has the discrete topology. Then the infinite product X with the product topology is also…

A: It is given that X has the discrete topology. Check whether the infinite product Xw with product…