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**1) Endow Z with the toplogy T consisting of the empty set and all subsets containing 0.**

Let g(Z,T)->(Z,T) be continouous and bijective. Prove that g is a homeomorphism.

Clearly I have to prove $g^{-1}$

Let g(Z,T)->(Z,T) be continouous and bijective. Prove that g is a homeomorphism.

Clearly I have to prove $g^{-1}$

**is continuous but I can't see how to.**

2) let f map polynomials to polynomials. Prove that if f(p(x))=1+1/2$\int_{0}\,^{x} p(t)\,dt$, the f is a contraction mapping with no fixed points

3)Let

2) let f map polynomials to polynomials. Prove that if f(p(x))=1+1/2$\int_{0}\,^{x} p(t)\,dt$, the f is a contraction mapping with no fixed points

3)

**B be a subset of X. For each B define the topolgy τ to consist of the subsets U of X such that U∩B is empty, plus the empty set. Let A be an infinite subset of X. Show that A is compact in X if and only if A ∩ B is not equal to the ∅.**