What is the minimum number of points that can be contained in a space

Q: Give and solve example of complete space

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Q: Find an open cover of the interval (−∞, 3]. Justify why that open cover has no finite subcover.

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

A: A set is said to be infinite countable if its elements can be counted like natural number but you…

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: Give an example of a set w-limit that is a minimal set.

A: To give an example of a set w-limit that is a minimal set.  Given a collection of sets , a member…

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: Find the transitive closure of R={(0,1),(0,2),(1,1), (1,3),(2,2),(3,0)}

A: Given relation is                              R=(0, 1), (0, 2), (1, 1), (1, 3), (2, 2), (3, 0) we…

Q: Show that recel projeetrue space is an manifol a.

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Q: Define a compact space.

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

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: %3D

A: 1. A =x ∈R2| 1≤∥x∥≤9    ∥x∥=x2+y2      1≤∥x∥≤9⇒1≤ x2+y2≤9⇒1≤x2+y2≤81 the graph of A is

Q: show that (0.1)with usual topology is connected

A: Sol

Q: Compute: Limit points. Interior of the set. Closure of the set. Boundary of the set of the set…

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Q: The boundary of a set is equal to the boundary of the complement) True or false

A: (The boundary of a set is equal to the boundary of the complement) True or false

Q: If X and Y are separable spaces, then X × Y is separable.

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Q: Each nonempty can be represented as the convex hull of its extreme points. --------

A: Each of the nonempty compact convex set can be represented as the convex hull of its extreme point.

Q: What is the locus of points in space that are equidistant from the vertices of equilateral triangle…

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

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

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Q: Show that the norm of the maximum satisfies the definition of norm

A: The objective is to show that the norm of the maximum satisfies the definition of norm. Consider the…

Q: What is the norm of a partition of a closed interval?

A: Concept: The calculus helps in understanding the changes between values that are related by a…

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

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Q: denotes the set of subsp

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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: what is the maximum size of independent set ? And give examples

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Q: Give two evamples that infinite intersection of open Sets not be open. may

A: See the attachment

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: Can you show an example of Null space?

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Q: How different can two sets of R ^ 2 that have the same boundary be?

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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: What is the Boundedness Theorem?

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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: SHOW that the continuous image Path - connected space is path Connected.

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Q: Show that closed ball in metric space (x,d) is a closed set. Also,

A: Given: (x,d) is a metric space The objective is to show that the closed ball y in (x,d) is a closed…

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

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Q: topological spaces X and Y,

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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: Every uncountable set in a 2nd countable space has a limit point.

A: We have to prove that every uncountable set in 2nd countable space a limit point. Definition:- Let…

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

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

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Q: Exhibit an open cover of the interval (1, 2] that has no finite subcover.

A: An open cover of A is a collection of open subsets of A whose union is all of A and that a finite…

Q: 5. Can somcone cross all the bridges shown in this map cxactly once and return to the starting…

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

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.