Topology Problems

1/ On the three point set X ={a,b,c}, the trivial topology has two open set and the discrete topology has eight open sets. For each of n=3,,….,7, either find a topology on X consisting of n open sets or prove that no such topology exist.

2/ Give an example of a space where the discrete topology is the same as the finite complement topology.

3/ Make and prove a conjecture indicating for what class of sets the discrete and finite complement topologies Scoincide.

4/ Define a topology on R by listing open sets within it that contains the open sets (0,1)and(1,3)and that contain and that contains as few open sets a few sets as possible.

5/
let X be a set and assume p∈X. Show that the collection τ consisting of X and all subsets of X that exclude p, is topology on X. This topology is called the excluded point topology on X and we denote it by EPXp

Let τ consist of ∅,R and all intervals(-∞,p) for p∈R. Prove that τ is a topology on R.

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