prove that the following is a basis for topology on the prescribed set. X = {p/

Question

prove that the following is a basis for topology on the prescribed set. X = {p/ p is a polynomial with real coefficients} and the collections of subsets of X of the form
Bn = {p belongs to X / degree of p = n} where n is a nonnegative integer.

Explanation / Answer

Your space is the collection of all polynomials with real coefficients. Take any element x in X. This is a polynomial with real coefficients of some degree n. This element x belongs to the base set Bn. Therefore the sets Bn cover X. Take any two base sets Ba and Bb and intersect them to form C. It is clear that either Ba is a subset of Bb or Bb is a subset of Ba, because polynomials of degree n are also polynomials of degree m, where m >= n (For example, x^2 is a polynomial of degree 3 when we consider 0*x^3 + 1*x^2 + 0*x + 0). Now C = Ba, or C = Bb. In either case, C is a base set. Then for any element in C, there is a base set which is a subset of C (C itself), which contains that element. QED

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