2  Continuity

2.1 Continuity

If \((X, \mathcal T)\) is any topological space and \(Y \subset X\), then \(\{U \cap Y : U \subset \mathcal T\}\) is easily seen to be a topology on \(Y\), called the relative topology.

Let \(f: A \to B\). Then for any \(C \subset B\), the inverse image of \(C\) under \(f\) is \(f^{-1}(C) := \{x \in A : f(x) \in C\}\). The inverse image preserves unions and intersections: for any non-empty collection \(\{B_i\}_{i\in I}\), \(B_i \subset B\):

  • \(f^{-1}(\bigcup_{i\in I}B_i) = \bigcup_{i \in I}f^{-1}(B_i)\)
  • \(f^{-1}(\bigcap_{i\in I}B_i) = \bigcap_{i \in I}f^{-1}(B_i)\)

When \(I\) is empty, the equation for union still holds, with both sides empty. The equation for intersections would be true if we define \(\bigcap_{i \in \varnothing} X_i = X\) for \(X = A\) or \(B\).

The notion of continuity really depends only on topology and can be defined as follows:

Definition 2.1 Given topological spaces \((X, \mathcal T)\) and \((Y, \mathcal U)\), a function \(f: X \to Y\) is called continuous iff \(\forall U \in \mathcal U\), \(f^{-1}(U) \in \mathcal T\).

Example 2.1 Consider \(f: \mathbb R \to\mathbb R\), \(f(x) = x^2\). Let \(U = (a, b)\). Then:

  • If \(b \le 0\), \(f^{-1}(U) = \varnothing\)
  • If \(a < 0 < b\), \(f^{-1}(U) = (-b^{1/2}, b^{1/2})\)
  • If \(0 \le a < b\), \(f^{-1}(U) = (-b^{1/2}, -a^{1/2})\cup (a^{1/2}, b^{1/2})\)

So the inverse image of an open interval under \(f\) is not always an interval, but is always an open set.

If \(n(\cdot): \mathbb N \to \mathbb N\) an increasing function, then for a sequence \(\{x_n\}\), the sequence \(k \mapsto x_{n(k)}\) will be called a subsequence of \(\{x_n\}\).

If \(x_n \to x\), then any subsequence \(k \mapsto x_{n_(k)}\) also converges to \(x\). If \(\mathcal T\) is the topology defined by a pseudometric \(d\), then it is easily seen that for any sequence \(x_n \in X\), \(x_n \to x\) iff \(d(x_n, x) \to 0\).

2.2 Nets and Filters

Converging along a sequence is not the only way to converge. A function \(f\) is continuous at \(x\) would means that \(f(y)\to f(x)\) as \(y \to x\). This implies that, for every sequence s.t. \(y_n \to x\), \(f(y_n)\to f(x)\). However, in some non-metrizable topological spaces, sequence are inadequate. It might happen, that for every \(x\) and every sequence \(y_n \to x\), we have \(f(y_n) \to f(x)\) but \(f\) is not continuous.

This gives rise to the two main convergence concepts, for “nets” and “filters” that generalize what sequences do in metric spaces but for general topological spaces.

Definition 2.2  

  • A directed set is a partially ordered set \((I, \le)\) s.t. for any \(i, j \in I\), there is a \(k \in I\) with \(k \ge i\) and \(k \ge j\). In other words, a directed set is a set with transitive, antisymmetric relation \(\le\) and every pair of elements has an upper bound.

Definition 2.3  

  • A net \(\{x_i\}_{i\in I}\) is any function \(x\) whose domain is a directed set, written \(x_i := x(i)\). In a topological space \((X, \mathcal T)\), a net \(\{x_i\}_{i\in I}\) converges to \(x \in X\), written \(x_i \to x\), iff for every neighborhood \(A\) of \(x\), there is a \(j \in I\) s.t. \(x_k \in A\) for all \(k \le j\).

Example 2.2  

  • A trivial directed set is the set of positive integers with usual ordering (since this is also a total ordering). For this directed set, a net is a sequence, thus sequences are a special case of nets.
  • Another example, let \(I\) be the set of all finite subsets of \(\mathbb N\), partially ordered by inclusion. Then if \(\{x_n\}_{n \in \mathbb N}\) is a sequence of real numbers, \(F \in I\), let \(S(F) = \sum_{n \in F} x_n\). Then \(\{S(F)\}_{F\in I}\) is a net. If it converges, the sum \(\sum_n x_n\) is said to converge unconditionally (this is equivalent to absolute convergence, \(\sum_n |x_n| < \infty\)).

Example 2.3 An example of nets is the Riemann integral.

Let \(a, b \in \mathbb R\), \(a < b\); \(f: [a, b] \to \mathbb R\). Let \(I\) be the set of all finite sequence \(a = x_0 \le y_1 \le x_1 \le y_2 \le x_2 \dots \le y_n \le x_n = b\), where \(n\) may be any positive integer. Write \(u := (x_j, y_j)_{j \le n}\) (\([x_j, x_{j+1}]\) are the sub-intervals, and \(y_j \in [x_{j-1}, x_j]\) are the sample points).

If also \(v\in I\), \(v = (w_i, z_j)_{j \le m}\), we define the ordering \(v \le u\) iff:

  • \(m < n\), and
  • For each \(j \le m\), there is an \(i \le n\) with \(x_i = w_j\).

This is to say that the partition \(\{x_0, ..., x_n\}\) of \([a, b]\) is a refinement of the partition \(\{w_0, ..., w_m\}\), keeping the \(w_j\) and inserting additional points. This ordering makes \(I\) a directed set, and does not invole the \(y_j\).

Let \(S(f, u) := \sum_{1 \le j \le n} f(y_j)(x_j - x_{j-1})\). This is a net. The Riemann integral of \(f\) from \(a\) to \(b\) is defined as the limit of this net iff it converges to some real number.

Definition 2.4  

  • Given a set \(X\), a filter base in \(X\) is a non-empty collection \(\mathcal F\) of non-empty subsets of \(X\) s.t. for any \(F, G \in \mathcal F\), \(F \cap G \supset H\) for some \(H \in \mathcal F\).
  • A filter base \(\mathcal F\) is called a filter iff whenever \(F \in \mathcal F\), and \(F \subset G \subset X\) then \(G \in \mathcal F\).

A filter \(\mathcal F\) can be equivalently defined as a non-empty collection of non-empty subsets of \(X\) such that:

  • \(F \in \mathcal F\) and \(F \subset G \subset X \implies G \in \mathcal F\)
  • \(F, G \in \mathcal F\implies F \cap G \in \mathcal F\)

If \(\mathcal F\) is any filter base, then \(\mathcal G = \{G \subset X: F \subset G \text{ for some } F \in \mathcal F\}\) is a filter; and \(\mathcal F\) is said to be a base of \(\mathcal G\).

Definition 2.5  

  • The filter base \(\mathcal F\) is said to converge to a point \(x\), written \(\mathcal F \to x\), iff every neighborhood of \(x\) belongs to the filter \(\mathcal G\).

That is,

  • The set of all neighborhoods of a point \(x\) is a filter converging to \(x\).
  • The set of all open neighborhoods of \(x\) is a filter base converging to \(x\).

Example 2.4 Given a continuous real function \(f: [0,1] \to \mathbb R\), let \(t := \sup\{f(x): 0 \le x \le 1\}\).

  • Let \(I_0 := [0,1]\). Then the supremum of \(f\) on at least one of \([0, 1/2]\) or \([1/2, 1]\) equals \(t\). Let \(I_1\) be such an interval of length \(1/2\).
  • We define \(\{I_n\}\) recursively: given a closed interval \(I_n\) of length \(1/2^n\) on which \(f\) has the same supremum \(t\) as on all of \([0,1]\), and \(I_{n+1}\) be a closed interval, either the left half or right half of \(I_n\) with the same supremum.

Then \(\{I_n\}_{n\ge 0}\) is a filter base converging to a point \(x\) for which \(f(x) = t\).

If \(X\) is a set, \(f\) is a function with \(\text{dom }f \supset X\), for each \(A \subset X\), define \(f[A] := \text{ran}(f|A) = \{f(x): x \in A\}\). For any filter base \(\mathcal F\) in \(X\) let \(f[[\mathcal F]] := \{f[A]: A \in \mathcal F\}\). Note that \(f[[\mathcal F]]\) is also a filter base.

Theorem 2.1 Given topological spaces \((X, \mathcal T)\) and \((Y, \mathcal U)\), and a function \(f: X \to Y\), the following are equivalent:

  • \((1)\) \(f\) is continuous
  • \((2)\) For every convergent net \(x_i \to x\) in \(X\), \(f(x_i)\to f(x)\) in \(Y\)
  • \((3)\) For every convergent filter base \(\mathcal F \to x\) in \(X\), \(f[[\mathcal F]] \to f(x)\) in \(Y\)
  • \((1) \implies (2)\): suppose \(f(x) \in U \in \mathcal U\). Then \(x \in f^{-1}(U)\), so for some \(j\), \(x_i \in f^{-1}(U)\) for all \(i > j\). Then \(f(x_i) \in U\), so \(f(x_i)\to f(x)\).
  • \((2) \implies (3)\): let \(\mathcal F \to x\). If \(f[[\mathcal F]] \not\to f(x)\), i.e. \(f[[\mathcal F]]\) does not converge to \(f(x)\), take \(f(x) \in U \in \mathcal U\) with \(f[A]\not\subset U\) for all \(A \in \mathcal F\). Define a partial ordering on \(\mathcal F\) by \(A\le B\) iff \(A \supset B\) for \(A\) and \(B\) in \(\mathcal F\). By definition of filter base, \((\mathcal F, \le)\) is then a directed set. Define a net by choosing, for each \(A \in \mathcal F\), an \(x(A) \in A\) with \(f(x(A)) \not\in U\). Then the net \(x(A)\to x\) but \(f(x(A))\not\to f(x)\), contradicting (2).
  • \((3) \implies (1)\): take any \(U \in \mathcal U\) and \(x \in f^{-1}(U)\). The filter \(\mathcal F\) of all neighborhoods of \(x\) converges to \(x\), so \(f[[\mathcal F]] \to f(x)\). For some neighborhood \(V\) of \(x\), \(f[V] \subset U\), so \(V \subset f^{-1}(U)\), and \(f^{-1}(U)\in \mathcal T\).

2.3 Interior and Closure

Definition 2.6 A topological space \((X, \mathcal T)\) is called a Hausdorff space iff for every two distinct points \(x, y \in X\), there are open sets \(U, V\) with \(x \in U\), \(y \in V\) and \(U \cap V = \varnothing\).

Thus a pseudometric space \((X, d)\) is Hausdorff if and only if \(d\) is a metric.

Definition 2.7 For any topological space \((S, \mathcal T)\) and a set \(A \subset S\), the interior of \(A\): \(\text{int }A := \bigcup \{U \in \mathcal T: U \subset A\}\) is the largest open set included in \(A\).

The closure of \(A\): \(\bar A := \bigcap\{F \subset S: F \supset A \text{ and } F \text{ is closed}\}\).

For any sets \(U_i \subset S\), for \(i \in I\), \(S \setminus (\bigcup_{i \in I} U_i) = \bigcap_{i \in I} (S \setminus U_i)\). Since any union of open sets is open, thus \(\bar A\) is the smallest closed set including \(A\).

Theorem 2.2 Let \((S, \mathcal T)\) be any topological space. Then:

  • \((1)\) For any \(A \subset S\), \(\bar A\) is the set of all \(x \in S\) s.t. some net \(x_i \to x\) with \(x_i \in A\) for all \(i\).
  • \((2)\) A set \(F \subset S\) is closed iff for every net \(x_i \to x\) in \(S\) with \(x_i \in F\) for all \(i\) we have \(x \in F\).
  • \((3)\) A set \(U \subset S\) is open iff for every \(x \in U\) and \(x_i \to x\) there is some \(j\) with \(x_i \in U\) for all \(i \ge j\).
  • \((4)\) If \(\mathcal T\) is metrizable, nets can be replaced by sequences \(x_n \to x\) in \((1)\), \((2)\), and \((3)\).
  • \((1)\) If \(x \not \in \bar A\) and \(x_i \to x\), then \(x_i \not \in A\) for some \(i\). Conversely, if \(x \in \bar A\), let \(\mathcal F\) be the filter of all neighborhoods of \(x\). Then for each \(N \in \mathcal F\), \(N \cap A \ne \varnothing\). Choose \(x(N) \in N \cap A\). Then the net \(x(N) \to x\).
  • \((2)\) Note that \(F\) is closed iff \(\bar F = F\), and apply \((1)\).
  • \((3)\) “Only if” follows from the definition of convergence of nets. “If”: suppose a set \(B\) is not open. Then for some \(x \in B\), by \((2)\) there is a net \(x_i \to x\) with \(x_i \not \in B\) for all \(i\).
  • \((4)\) In the proof of \((1)\) we can take the filter base of neighborhood \(N = \{y: d(x, y) < 1/n\}\) to get a sequence \(x_n \to x\). The rest follows.

Definition 2.8  

  • For any topological space \((S, \mathcal T)\), a set \(A \subset S\) is said to be dense in \(S\) iff the closure \(\bar A = S\).
  • \((S, \mathcal T)\) is said to be separable iff \(S\) has a countable dense subset.

Example 2.5 The set \(\mathbb Q\) of all rational numbers is dense in the line \(\mathbb R\), so \(\mathbb R\) is separable (for the usual metric).

Definition 2.9  

  • \((S, \mathcal T)\) is said to satisfy the first axiom of countability, or to be first-countable, iff there is a countable neighborhood-base at each point.
  • \((S, \mathcal T)\) is said to satisfy the second axiom of countability, or to be second-countable, iff \(\mathcal T\) has a countable neighborhood-base. Clearly any second-countable space is also first-countable.

Example 2.6  

  • For any pseudometric space \((S, d)\), the topology is first-countable, since for each \(x \in S\), \(B(x, 1/n)\) forms a neighborhood-base at \(x\).

Proposition 2.1 A metric space \((S, d)\) is second-countable iff it is separable.

  • \(\Leftarrow\): Since \(S\) is separable, let \(A\subset S\) be countable and dense. Let \(\mathcal U\) be the set of all balls \(B(x, 1/n)\) for \(x \in A\) and \(n = 1,2,...\). To show that \(\mathcal U\) is a base, let \(U\) be any open set, \(y \in U\); then for some \(m\), \(B(y, 1/m) \subset U\). Take \(x \in A\) with \(d(x, y) < 1/(2m)\). Then \(y \in B(x, 1/(2m)) \subset B(y, 1/m) \subset U\), so \(U\) is the union of the elements of \(\mathcal U\) that it includes, and \(\mathcal U\) is a countable base.
  • \(\Rightarrow\): Suppose there is a countable base \(\mathcal V\) for the topology, which we may assume consists of non-empty sets. By the axiom of choice, let \(f\) be a function on \(\mathbb N\) whose range contains at least one point of each set in \(\mathcal V\). Then this range is dense.