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 subsetU⊂Ris anopen subsetif for everyx∈Uthere is an open interval (a, b) such thatx∈(a, b) and (a, b)⊂U.Letf:R→Rbe a function.Prove that the following statements areequivalent.•For every open setVofR, the pre-imagef-1(V) is an open subsetofR.•For everyx0∈Rand for every open setVcontainingf(x0), there isan open setUcontainingx0such thatf(U)⊂V.•For everyx0∈Rand 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.