Tutorial 1 - MATH4300 General Topology Problem set 1 1(Intermediate Value Theorem Let p(x be a polynomial with integer coecients of odd degree Show that

Tutorial 1 - MATH4300 General Topology Problem set 1...

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MATH4300. General Topology. Problem set 1. 1. (Intermediate Value Theorem) Letp(x) be a polynomial with integer co-efficients of odd degree. Show that the equationp(x) = 0 has at least onereal solution.2. ( -δContinuity) Recall that a subsetURis anopen subsetif for everyxUthere is an open interval (a, b) such thatx(a, b) and (a, b)U.Letf:RRbe a function.Prove that the following statements areequivalent.For every open setVofR, the pre-imagef-1(V) is an open subsetofR.For everyx0Rand for every open setVcontainingf(x0), there isan open setUcontainingx0such thatf(U)V.For everyx0Rand for every>0, there isδ >0 such that if|x-x0|< δthen|f(x)-f(x0)|<.3. (Basis for a topology ofR2) Page 92, problem 6.4. (Bijections) For anyα < βwhereα,βare reals,or-∞, constructexplicit bijections between the following sets of the real numbers.

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