A Course in Point Set Topology by John B. Conway

Author:John B. Conway
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

The proof of the next proposition is straightforward (Exercise 2) and a good opportunity to fix the ideas in your mind.

Proposition 2.7.7.

(a) If a net in a topological spaces converges to x, then it clusters at x.

(b) A net can converge to only one point.

Here is one way to define a subnet that extends the idea of a subsequence. If I is a directed set, say that a subset J is cofinal if for every i in I there is a j in J with j ≥ i. It is easily shown that when J is a cofinal subset, then it too is a directed set. So if we have a net , then we could define a subnet to be the restriction of x to some cofinal subset. However, this is NOT the definition of a subnet. The more complicated notion of a subnet is formulated in such a way that various results about sequences and subsequences in a metric space can be extended to nets and subnets in a topological space. For example: if a net clusters at x, then it has a subnet that converges to x; a topological space X is compact if and only if every net in X has a convergent subnet.

The interested reader can consult [4] and [6] for the accepted definition of a subnet. We will not use subnets in this book. I have found that I can usually avoid them and have decided to live my life that way. I am not saying that all the readers should follow my example, but for this introduction we will follow the path of least resistance.

Proposition 2.7.8.

Let X and Z be topological spaces.

(a) If and x ∈ X, then f is continuous at x if and only if whenever in X, in Z.

(b) If is continuous at x and , then .

(c) A subset F of X is closed if and only if whenever we have a net {x i } of points in F that converges to a point x, we have that x ∈ F.

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