Lecture Notes Topology (2301631) Phichet Chaoha Department of ...

8 Topology (2301631)
Exercise 1.6. In R with the cofinite topology, prove that any two nonempty
open sets always intersect. Is this still true for R equipped with the cocountable
topology?
Exercise 1.7. For a collection {τ α } α∈Λ of topologies on a set X, show that
the intersection ∩ α∈Λ τ α is still a topology on X. What about ∪ α∈Λ τ α?
Definition 1.8. A subset A of a space X is said to be
• closed if A c = X − A is an open set,
• G δ if A is a countable intersection of open sets,
• F σ if A is a countable union of closed sets, or equivalently, if A c is a G δ -set
in X.
Example 1.9. In R (with all usual open subsets), Q = ∪ q∈Q {q} is clearly F σ,
and hence Q c is G δ . However, both of them are neither open nor closed.
Definition 1.10. Let (X, τ X ) be a space, x ∈ X and A ⊆ X. A subset U of
X is called a neighborhood of x if x ∈ U ∈ τ X . Similarly, a subset V of X is called
a neighborhood of A if A ⊆ V ∈ τ X .
Definition 1.11. A space X is said to be Hausdorff if any two distinct points
of X have disjoint neighborhoods.