4. (a)
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State and prove Cantor's intersection theorem.
Answers
Answer:
Cantor's intersection theorem refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty compact sets.
Answer:
The Cantor set is defined as the crossroad of a dwindling nested sequence of sets, each of which is defined as the union of a finite number of unrestricted intervals; hence, each of these sets isnon-empty, unrestricted, and bounded. This description is a straight forward corollary of the theorem.
In order to demonstrate this, we will first demonstrate that| A|| P( A)| and also that( 2)| A| = | P( A)|. This is original to the expression of the handed theorem's strict lower than clause.( 1)| A| ≤| P( A)| Now, to demonstrate this, all we've to do is produce a bijection between A and a subset ofP.( A). also we'll know that group, which can not be bigger than P, has the same size asA.( A). Consider the collection of all A's single- element subsets, E = x x A. Since it's composed of several subsets of A, it's apparent that E P( A). It's also a licit subset because we're apprehensive that it's devoidof.Thus its is proofed.
To know more about intersection
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To know more about Cantor's theorem
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