Download TOPOLOGY WEEK 5 Proposition 0.1. Let (X, τ) be a topological

TOPOLOGY WEEK 5
Proposition 0.1. Let (X, τ ) be a topological space. A sequence (xn )n∈N converges to x ∈ X
if and only if for every U ∈ τ such that x ∈ U ∃N ∈ N such that for every n ≥ N xn ∈ U .
(equivalently, for every U ∈ τ such that x ∈ U there exists all but finitely many xn ∈ U . )
Definition 0.1. Let (X, τ ) be a topological space. A subset F ⊆ X is sequentially closed if
for every convergent sequence (xn )n∈N ⊆ F , its limit is contained in F .
Definition 0.2. A topological space (X, τ ) is called sequential if every sequentially closed
subset of X is closed.
(1) Assume (X, τ ) is a first countable topological space
(a) For any E ⊆ X, x ∈ E if and only if there is a sequence (xn )n∈N contained in E
which converges to x.
(b) Prove that every first countable topological space is sequential. (Note: All metric
spaces are first countable and therefore sequential).
(2) Prove that if (X, τX ) and (Y, τY ) are sequential, then f ∶ X → Y is continuous if and
only if whenever xn → x in X, then f (xn ) → f (x) in Y .
(3) This problem gives an example of sequential topological space that is not first countable. Let (X, τcf ) be an uncountable set with the cofinite topology.
(a) Prove that (X, τcf ) is not first countable. (This is the same proof that an uncountable set with the cocountable topology is not first countable).
(b) Prove that any sequence of distinct points converges to every point in X.
(c) Prove that (X, τcf ) is sequential.
(4) This problem gives an example of a space that is not sequential and not first countable.
Let (X, τcc ) be an uncountable set with the cocountable topology.
(a) Prove that every convergent sequence is eventually constant.
(b) Find a set that is sequentially closed but not closed. (Hint: Characterize the
closed sets).
Date: April 22, 2015.
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