Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

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: i. PROVE THAT THE EXCLUDED POINT TOPOLOGY ON ANY SET WITH AT LEAST 2 POINTS IS NOT HAUSDORFF. ii.…

A: Definition: A topological space X, τ is said to be Hausdorff if for every x, y∈X there exits U, V∈τ…

Q: Find an example of a nonempty set satisfying: • Open, every point is a limit point • Closed, every…

A: Just because a nonempty open set contains only limit points does not mean that the set contains all…

Q: show that any infinite discrete topological space is not compact

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Q: let A be a bounded. Sup A non. n-empty set in R show that is a limit point of A.

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Q: 2- Every finite set in any topological space is a- Regular set, b- connected set, c- Hausdroff' set,…

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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: Provide a direct proof that a sequentially compact set must be both closed and bounded without…

A: WE GIVE A DIRECT PROOF. First, we prove that set A is closed then A is bounded.

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 that a nonempty closed subset of R, if it is bounded from below, has a least element.

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Q: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

A: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

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: Show that any finite union of Nowhere Dense Sets is Nowhere Dense.

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Q: A relation ''<'' is 1) İrreflexive. 2) Transitive. called a quasi-order on a set S if its: صواب ihi

A: We know that

Q: Prove that if S is a bounded set then S is bounded.

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Q: Set is closed

A: To prove that the Cantor Set is closed.

Q: Prove that T is the discrete topology for X iff every point in X is an open set

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Q: Exercise 2.7. Give an example of a topological space and a collection of open sets in that…

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Q: Give an example of a bounded set that has infinitely many isolated points

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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: 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: Prove that the Cantor set has measure zero.

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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 the Cantor Set contains no open intervals

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Q: Tet X and Y be infinite sets prove that XUY is infinite

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Q: Show that the lower limit topology on R is strictly finer than the standard topology on R. That is:…

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Q: Prove that interval [0,1] is uncountable.

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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: Question No 1 (8). Prove that the discrete topology on an uncountable set does not Satisty the…

A: Note- Let X be a topological space, then X satisfies the second axiom of countability or is second…

Q: Let ACR be a nonempty bounded set. Prove that sup A is unique.

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Q: Set A is bounded below if and only if set -A is bounded above. True False

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Q: i) show that a closed interval is a closed set;

A: Note: According to bartleby we have to do only first question please upload the question separately.…

Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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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: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

Q: What is the Boundedness Theorem?

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Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

A: Compact Set: A collection A of subsets of space X is said to be cover X or covering of X if the…

Q: Prove that the smallest member of a set, if exits, is the greatest lower bound of the set. Correct…

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Q: Prove or disprove: All uncountable measurable subsets of ℝ have positive measure.

A: All uncountable measurable subsets of ℝ have positive measure.

Q: Show that the interval [0,1] is a closed set in R

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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: Let T be the finite closed topology and be also the discrete topology on a non empty set X. Then…

A: X is finite

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

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Q: Using the definition of an open set, prove that the interval (0, 1) is an open set.

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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: Give example closed set and bounded set is compact

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Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

Q: 28

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Q: Define bounded set and an example of a set which is bounded below but not bounded above.

A: Bounded set:  A set of real numbers is said to be bounded set if and only if it has an upper and…