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

22 Topology (2301631)
Definition 1.88. A directed set is a preordered set (I, ≤) such that for any
α, β ∈ I, we have α ≤ γ and β ≤ γ for some γ ∈ I.
Example 1.89. Clearly, (P(S), ⊆) and (N, |) are directed sets. Moreover, if
N (x) denotes the set of all neighborhoods of x ∈ X, both (N (x), ⊆) and (N (x), ⊇)
are also directed sets.
Definition 1.90. A net in a space X is simply a map (x α ) : I → X where I
is a directed set.
Example 1.91. A sequence in a space X is simply a net in X whose directed
set is (N, ≤).
Definition 1.92. A net (x α ) : I → X is said to converge to x ∈ X, abbreviated
by (x α ) → x, if for each neighborhood U of x, there exists γ ∈ I such that x α ∈ U
for all α ≥ γ.
Exercise 1.93. Prove that a convergent net in a Hausdorff space has a unique
limit.
Theorem 1.94. Let X be a space, x ∈ X and A ⊆ X. There is a net in A
converging to x iff x ∈ A.
Proof. (⇒) Let (a α ) : (I, ≤) → A be a net in A converging to x, and U a
neighborhood of x. Then, by the definition of convergence, there exists γ ∈ I such
that a α ∈ U for all α ≥ γ. In particular, a γ ∈ U ∩ A. Hence, U ∩ A ≠ ∅; i.e, x ∈ A.
(⇐) Suppose x ∈ A. Let I be the poset of neighborhoods of x partially ordered
by inverse inclusions; i.e., U ≤ V iff U ⊇ V . Then for each U ∈ I, we have U∩A ≠ ∅.
By the axiom of choice, we can form a net (a U ) in A such that a U ∈ U ∩ A for each
U ∈ I. Now, it is clear from the construction that (a U ) → x.
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Exercise 1.95. Let X and Y be spaces, x ∈ X and f : X → Y a map.
(1) Prove that if f is continuous at x, then for each sequence (x n ) in X
converging to x, the sequence (f(x n )) converges to f(x).
(2) Show that the converse of the above statement fails in general.
(3) Prove that a map f : X → Y is continuous at x iff for each net (x α ) in X
converging to x, the net (f(x α )) converges to f(x).
Definition 1.96. A subset I ′ of a directed set I is said to be cofinal in I if
for each α ∈ I, we have α ≤ α ′ for some α ′ ∈ I ′ .
Proposition 1.97. A cofinal subset of a directed set is also a directed set.
Proof. Suppose I ′ be a cofinal subset of a directed set I. Let α ′ , β ′ ∈ I ′ .
Since I ′ ⊆ I and I is directed, we clearly have α ′ ≤ γ and β ′ ≤ γ for some γ ∈ I.
Since I ′ is cofinal in I, there exists γ ′ ∈ I ′ such that γ ≤ γ ′ . Hence, α ′ ≤ γ ′ and
β ′ ≤ γ ′ as required.
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Definition 1.98. A net (y β ) : J → X is said to be a subnet of a net (x α ) :
I → X if (y β ) = (x α ) ◦ f for some order-preserving map f : J → I where f(J ) is
cofinal in I.
Example 1.99. A subsequence is clearly a subnet.
Exercise 1.100. Let X be a Hausdorff space, (x α ) : I → X a convergent net
in X and (y β ) : J → X a subnet of (x α ). Prove that if (y β ) converges to a point
z ∈ X, then so does (x α ).