Theorem: A set in R is compact if and only if it is closed and bounded. wwwwwwwwwww A) Determine and justify whether the set [-1, 1] is compact or not. B) Justify because the function f(x)=x^2 is continuous, also find the image of [-1, 1] under f, wwwwwwwwwwwwwwwwmmmmm wwwwwwww that is, f([-1, 1]), justify because f([-1, 1]) is compact.
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![Theorem: A set in R is compact if and only if it is closed and bounded.
A) Determine and justify whether the set [-1, 1] is compact or not.
B) Justify because the function f(x)=x^2 is continuous, also find the image of [-1, 1] under f,
that is, f([-1, 1]), justify because f([-1, 1]) is compact.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7536a213-9eec-4280-9405-2460faef03dd%2Fd075117c-6d6e-4148-870c-f1aa8a0bb2b6%2Fv3xh4uf_processed.jpeg&w=3840&q=75)
What is Compact Set:
Compactness is a quality used in mathematics, specifically general topology, to more precisely define the idea of a space having no gaps or missing endpoints, i.e., that the space does not exclude any limiting values of points. A topological space is said to be sequentially compact if every infinite sequence of points sampled from the space has an endless sub-sequence that converges to some point in the space, according to one such generalisation. A subset of Euclidean space is said to be compact in this sequential meaning by the Bolzano-Weierstrass theorem if and only if it is closed and bounded.
Given:
Given set is . Also, a function is given.
To Determine:
We verify whether the given set is compact or not. We also verify whether the image of is compact or not.
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