4  Product Topology

4.1 Ultrafilter

We consider the concept of an ultrafilter to motivate for the proof of an important theorem.

Definition 4.1 A filter \(\mathcal F\) in a set \(X\) is called an ultrafilter iff for all \(Y \subset X\), either \(Y \in \mathcal F\) or \(X \setminus Y \in \mathcal F\).

Example 4.1  

  • The simplest ultrafilters are of the form \(\{A \subset X : x \in A\}\) for \(x \in X\). These are called point ultrafilters.
  • The existence of non-point ultrafilters depends on the axiom of choice. Some filters converging to a point \(x\) are included in the point ultrafilter of all sets containing \(x\); but \((0, 1/n)\), \(n = 1, 2, ...\), for example, is a base of a filter converging to \(0\) in \(\mathbb R\), where no set in the base contains \(0\).

The next two theorems provide an analogue of the fact that every sequence in a compact space has a convergent subsequence.

Theorem 4.1 Every filter \(\mathcal F\) in a set \(X\) is included in some ultrafilter. \(\mathcal F\) is an ultrafilter iff it is maximal for inclusion, i.e. \(\mathcal F \subset \mathcal G\) and \(\mathcal G\) is a filter, then \(\mathcal F = \mathcal G\).

Theorem 4.2 A topological space \((S, \mathcal T)\) is compact iff every ultrafilter in \(S\) converges.

Definition 4.2 Given a topological space \((S, \mathcal T)\), a subcollection \(\mathcal U \subset \mathcal T\) is called a subbase for \(\mathcal T\) iff the collection of all finite intersections of sets in \(\mathcal U\) is a base for \(\mathcal T\).

Example 4.2 In \(\mathbb R\), a subbase of the usual topology is given by the open half-lines \((-\infty, b) := \{x:x < b\}\) and \((a, \infty) := \{x: x > a\}\).

Theorem 4.3 For any set \(X\) and collection \(\mathcal U\) of subsets of \(X\), there is a smallest topology \(\mathcal T\) including \(\mathcal U\), and \(\mathcal U\) is a subbase of \(\mathcal T\). Given a topology \(\mathcal T\) and \(\mathcal U \subset \mathcal T\), \(\mathcal U\) is a subbase for \(\mathcal T\) iff \(\mathcal T\) is the smallest topology including \(\mathcal U\).

Corollary 4.1  

  • If \((S, \mathcal V)\) and \((X, \mathcal T)\) are topological spaces, \(\mathcal U\) is a subbase of \(\mathcal T\), and \(f: S \to X\), then \(f\) is continuous iff \(f^{-1}(U) \in \mathcal V\) for each \(U \in \mathcal U\).
  • If \(S\) and \(I\) are any sets, and for each \(i \in I\), \(f_i: S \to X_i\) where \((X_i, \mathcal T_i)\) is a topological space, then there is a smallest topology \(\mathcal T\) on \(S\) for which every \(f_i\) is continuous. Here a subbase of \(\mathcal T\) is given by \(\{f_i^{-1}(U): i \in I, U \in \mathcal T_i\}\), and a base by finite intersections of such sets for different values of \(i\), where each \(\mathcal T_i\) can be replaced by a subbase of itself.

Example 4.3 This corollary can simplify the proof that a function is continuous. If \(f\) has real values, then, using a subbase for the topology of \(\mathbb R\) mentioned above, it is enough to show that \(f^{-1}((a, \infty))\) and \(f^{-1}((-\infty, b))\) are open for any real \(a, b\).

4.2 Product Topology

Definition 4.3 Let \((X_i, \mathcal T_i)\) be topological spaecs for \(i \in I\). Let \(X := \prod_{i\in I}X_i\), i.e. the set of all indexed families \(\{x_i\}_{i \in I}\) where \(x_i \in X_i\) for all \(i\). Let \(p_i\) be the projection from \(X\) onto the \(i\)th coordinate space \(X_i: p_i(\{x_j\}_{j \in I}) := x_i\) for any \(i \in I\). Let \(f_i = p_i\) gives a topology \(\mathcal T\) on \(X\), called the product topology, the smallest topology making all the coordinate projections continuous.

Example 4.4 Let \(\mathbb R^k := \{x = (x_1, ..., x_k): x_j \in \mathbb R \text{ for all } j\}\) be the Cartesian product of \(k\) copies of \(\mathbb R\) with product topology. The ordered \(k\)-tuple \((x_1, ..., x_k)\) can be defined as a function from \(\{1, 2, ..., k\}\) into \(\mathbb R\). We also write \(x = \{x_j\}_{1\le j \le k}= \{x_j\}_{j=1}^k\). The product topology on \(\mathbb R^k\) is metrized by the Euclidean distance.

For any real \(M > 0\), the interval \([-M, M]\) is compact in \(\mathbb R\). In \(\mathbb R^k\), the cube \([-M, M]^k\) is compact for the product topology is a special case of the following general theorem:

Theorem 4.4 (Tychonoff’s Theorem): Let \((K_i, \mathcal T_i)\) be a compact topological spaces for each \(i \in I\). Then the Cartesian product \(\prod_i K_i\) with product topology is compact.

Let \(\mathcal U\) be an ultrafilter \(\prod_i K_i\). Then for all \(i\), \(p_i[[\mathcal U]]\) is an ultrafilter \(K_i\), since for each set \(A \subset K_i\), either \(p_i^{-1}(A)\) or its complement \(p_i^{-1}(K_i \setminus A)\) is in \(\mathcal U\). Thus \(p_i[[\mathcal U]]\) converges to some \(x_i \in K_i\). For any neighborhood \(U\) of \(x := \{x_i\}_{i \in I}\) by definition of product topology, there is a finite \(F \in I\) and \(U_i \in \mathcal T\) for \(i \in F\) s.t. \(x \in \bigcap \{p_i^{-1}(U_i) : i \in F\} \subset U\). For each \(i \in F\), \(p_i^{-1}(U_i)\in \mathcal U\), so \(U \in \mathcal U\) and \(\mathcal U \to x\). So every ultrafilter converges, and \(\prod_i K_i\) is compact.

Among compact spaces, those which are Hausdorff spaces have especially good properties and are the most studied.

Proposition 4.1 Any compact set \(K\) in a Hausdorff space is closed.

On any set \(S\), the indiscrete topology is the smallest topology, \(\{\varnothing, S\}\). All subsets of \(S\) are compact, but only \(\varnothing\) and \(S\) are closed. This is the reverse of the usual situation in Hausdorff spaces.

Definition 4.4 If \(f: X \to Y\), \(g: Y\to Z\), let \((g \circ f)(x) := g(f(x))\) for all \(x \in X\). Then, \((g\circ f): X \to Z\) is called the composition of \(g\) and \(f\).

For any set \(A \subset Z\), \((g\circ f)^{-1}(A) = f^{-1}(g^{-1}(A))\). Thus, we have the following theorem.

Theorem 4.5 If \((X, \mathcal S)\), \((Y, \mathcal T)\), and \((Z, \mathcal U)\). If \(f\) is continuous from \(X \to Y\), \(y\) is continuous from \(Y \to Z\), then \(g \circ f\) is continuous from \(X \to Z\).

This is clear from the formulation of continuity in terms of convergent nets. If \(x_i \to x\), then \(f(x_i) \to f(x)\), so \(g(f(x_i)) \to g(f(x))\).

Definition 4.5 A homeomorphism of \((X, \mathcal S)\) onto \((Y, \mathcal T)\) is a 1-1 function \(f: X \to Y\) such that \(f\) and \(f^{-1}\) are continuous. If such an \(f\) exists, \((X, \mathcal S)\) and \((Y, \mathcal T)\) are called homeomorphic.

Example 4.5  

  • \((a, b)\) is homeomorphic to \((0, 1)\) by \(f(x) := a + (b-a)x\).
  • \((-1, 1)\) is homeomorphic to all of \(\mathcal R\) by \(f(x) := \tan (\pi x / 2)\).

Remark 4.1.

  • If \(f \circ h\) is continuous, \(h\) is continuous, \(f\) is not necessarily continuous. For example, \(h\) could be constant and \(f\) could be an arbitrary function, then \(f \circ h\) would be constant anyway.
  • If \(\mathcal T = 2^X\) the discrete topology on \(X\), \(h: X \to Y\) and \(f\) a function from \(Y\) into another topological space, then \(h\) and \(f \circ h\) are always continuous, but \(f\) can be arbitrary.

The following theorem will give the conditions to the continuity of \(f\).

Theorem 4.6 Let \(h\) be a continuous function from a compact topological space \(T\) onto a Hausdorff topological space \(K\). Then

  • A set \(A \subset K\) is open iff \(h^{-1}(A)\) is open in \(T\).
  • If \(f\) is a function from \(K\) into another topological space \(S\), then \(f\) is continuous iff \(f \circ h\) is continuous.
  • If \(h\) is 1-1, it is a homeomorphism.
  • Note that \(K\) is compact. Let \(h^{-1}(A)\) be open. Then \(T \setminus h^{-1}(A)\) is closed and hence compact. Thus \(h[T\setminus h^{-1}(A)] = K \setminus A\) is compact, hence closed, so \(A\) is open.
  • If \(f \circ h\) is continuous, then for any open \(U \subset S\), \((f\circ h)^{-1}(U) = h^{-1}(f^{-1}(U))\) is open. So \(f^{-1}(U)\) is open and \(f\) is continuous. The other implications are immediate from the definitions and the above theorem.