Give an example of a set in R ^ 2 with the usual topology that is not connected but has a connected boundary.

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

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Q: Find a star in the radial plane topology that does not contain any rationals.

A: The objective is to find the star in the radial plane topology which does not contain any rationals

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

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Q: Please define these cocepts: Isolated point, .discrete topology, dense space

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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: Every Hausdorff space is hereditarily Hausdorff.

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

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

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

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

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

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Q: State the Squeeze Theorem carefully.

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

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

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

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Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

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: attern on the graph as well as the space provided

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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: There are three conditions in which limits do not exist

A: First Condition   Limit fails to exist  when  The function approaches to some number from the right…

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

A:

Q: Give an example of a countable collection of bounded sets whose union bounded, and an example where…

A: Let A1 and A2  are bounded sets

Q: List the 4 types of indeterminate formsof limits and explain them

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Q: How do I write which intertvals are continuous for this graph?

A: NOTE: Refresh your page if you can't see any equations. . as we can see there is a jump at  and  and…

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: Show that an open subcover R is not compact by finding cover that does not have finite

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Q: Give an example that satisfies in arbitrary metric space, boundedness and closedness do not…

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Q: Can you show an example of Null space?

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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: Give an example of a norm space.

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

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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,

A:

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>0 is…

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: ) Find an example of an infinite collection of closed sets whose union is not closed.

A: Any closed interval is closed in R. From that we created this example.

Q: Give an example of Partition set based on your real life situation and describe the disjoint subset…

A: Partition set: Partion sets of a set are non-empty subsets that consists of the elements of the set…

Q: When Can Pointwise Converqance preserve boundedness?

A: To prove -  When can point wise convergence preserve boundedness.

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: Let (X, τ ) be a discrete space. Is (X, τ ) second countable? Explain why

A: Consider the provided question, Let x,τ be a discrete space,    ⇒x,τ  be denote topology if x,τ is…

Q: how that the union of a finite. number of bounded sets is bounded

A: A statement is given as, 'the union of a finite number of bounded sets is bounded. The objective is…

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

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Q: Show that the image of an open set under a continuous mapping need not be open

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Q: Give example closed set and bounded set is compact

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Q: Explain why a limit of the form 0∞ is not an indeterminate form.

A: The only indeterminate forms are: 0/0 ∞/∞ 0 · ∞ 00 ∞0 1∞ ∞ - ∞

Q: Construct an open cover for the interval [0, 1]. Verify that the open cover is in fact an open…

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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.