 # Topology Problems

## Question

Question 1:
Prove that the function max: R × R ---> R,   (x, y) ---> max{x,y} is continuous, where R × R have their Euclidean metrics.

Question 2:
Find a continuous bijection between topological spaces whose inverse is not continuous.

Question 3:
Let (X, T) be a topological space and take A ⊆ X.
Prove that there is a maximal open subset of X disjoint from A.
This is the exterior of A, denoted ext(A).

Question 4:
Let R[t] denote the set of all real polynomials in the intermediate t.
Let S be a subset of R[t].
Define V(S) := {x ∈ R | f(x) = 0 for every f ∈ S}.
Define F := {V(S) | S ⊆ R[t]}.
Prove that there is a topology on R, whose closed sets are precisely the elements of F and that this topology is not metrisable.
This is the Zariski topology on R and arises in algebraic geometry.

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