1  Topologies and Metrics

General topologies has to do with the notion of convergence. One of the main ways of defining convergence is by a metric, or distance \(d\), which is nonnegative and real-valued, with \(x_n \to x\) meaning \(d(x_n,x) \to 0\). The usual metric for real numbers is \(d(x,y) = |x-y|\). For the usual convergence of real numbers, a function f is called continuous if whenever \(x_n \to x\) in its domain, we have \(f(x_n) \to f(x)\).

Some interesting kinds of convergence are not defined by metrics: if we define convergence of a sequence of functions \(f_n\) “pointwise”, so that \(f_n \to f\) means \(f_n(x) \to f(x)\) for all \(x\), it turns out that there may be no metric \(e\) such that \(f_n \to f\) is equivalent to \(e(f_n, f) \to 0\).

Given a sense of convergence, we can call a set \(F\) closed if whenever \(x_i \in F\) for all \(i\) and \(x_i \to x\) we have \(x \in F\) also. Any closed interval \([a,b] := \{x: a \le x \le b\}\) is an example of a closed set. The properties of closed sets \(F\) and their complements \(U := X \setminus F\), which are called open sets, turn out to provide the best and most accepted way of extending the notions of convergence, continuity, and so forth to nonmetric situations.

1.1 Topologies

Definition 1.1 Given a set \(X\), a topology on \(X\) is a collection \(\mathcal T \subset 2^X\) of subsets of \(X\) such that:

• \(\varnothing \in \mathcal T\) and \(X \in \mathcal T\)
• Closed under finite intersections: \(U, V \in \mathcal T\) implies \(U \cap V \in \mathcal T\)
• Closed under arbitrary unions: \(\bigcup U \in \mathcal T\) if \(U \subset \mathcal T\)

The members of a topology are called open sets; and their complements, i.e. \(F = X\setminus U, U \in \mathcal T\) are called closed sets.

The pair \((X, \mathcal T)\) is called a topological space.

Example 1.1 Usual topology on \(\mathbb R\):

• Basic open sets are open intervals \((a, b) = \{x \in \mathbb{R} : a < x < b\}\).
• Open sets are arbitrary unions of such intervals.
• Closed intervals \([a, b] = \{x : a \leq x \leq b\}\) and single points.
• Half-open intervals \([a, b)\) and \((a, b]\) are neither open nor closed.

1.2 Metrics and Pseudometrics

Definition 1.2 A pseudometric \(d: X \times X \to \mathbb{R}_{+}\) satisfies:

• \(d(x, x) = 0\)
• Symmetry: \(d(x, y) = d(y, x)\)
• Triangle inequality: \(d(x, z) \leq d(x, y) + d(y, z)\)

A metric satisfies an additional property:

• \(d(x, y) = 0 \implies x = y\)

Example 1.2 The most common metrics:

• Usual metric on \(\mathbb R\): \(d(x,y):= |x-y|\)
• Usual metric on \(\mathbb R^n\), i.e. the Euclidean distance: \(d(x, y):= \sqrt{\sum_{i}(x_i - y_i)^2}\)

An example of a pseudometric which is not a metric over the space of Riemann-integrable functions over \([a,b]\): \[d(f, g) = \int_a^b |f(x) - g(x)| dx\] Here, \(d(f,g)=0\) does not imply \(f=g\).

1.3 Base, Neighborhood and Neighborhood-Base

Definition 1.3 A base for a topology \(\mathcal T\) is any collection \(\mathcal U \subset \mathcal T\) such that for every \(V \in \mathcal T\), \(V = \bigcup\{U \in \mathcal U : U \subset V\}\)

Since defining the topology \(\mathcal T\) often can be overwhelming, the concept of a base for a topology arises. A base is like a “starter kit” to build all open sets in the topology; in more rigorous terms, the base is a collection of open sets such that any open sets in the topology can be described as the union of sets in the base.

Definition 1.4  

• A neighborhood of a point \(x\) is any set \(N\) (open or not) such that \(x \in U \subset N\) for some open \(U\)
• A collection \(\mathcal N\) of neighborhood of \(x\) is a neighborhood-base at \(x\) iff for every neighborhood \(V\) of \(x\), \(x \in N \subset V\) for some \(N \in \mathcal N\)

A neighborhood-base at \(x\) is like a “minimal set of essential neighborhoods” from which you can describe any neighborhood of \(x\).

Example 1.3  

• The set of intervals \((x - 1/n, x + 1/n)\) for \(n = 1, 2, \dots\) forms a neighborhood-base at \(x\). At \(x = 0\), intervals \([-1/3, 1/4]\) and \([-1/8, 1/7]\) are neighborhoods of 0.

In a pseudometric space \((X, d)\), define the open ball centered at \(x\) with radius \(r\) to be \(B(x, r) := \{y \in X : d(x, y) < r\}\).

Theorem 1.1 For any pseudometric space \((X,d)\):

• The collection of all open balls \(B(x,r)\), for all \(x \in X\) and \(r > 0\), forms a base for a topology on \(X\).
• For fixed \(x \in X\), the collection of open balls centered at \(x\) forms a neighborhood-base at \(x\) for this topology.

Proof

We first show that any open set \(U\) in this topology can be written as a union of open balls.

Suppose \(x \in X\), \(y \in X\), \(r > 0\), and \(s > 0\).

Let \(U = B(x,r) \cap B(y,s)\). Suppose \(z \in U\). Let \(t := \min(r − d(x,z),s − d(y,z))\). Then \(t > 0\). To show \(B(z,t) \subset U\), suppose \(d(z,w) < t\). Then the triangle inequality gives \(d(x,w) < d(x,z) + t < r\). Likewise, \(d(y,w) < s\). So \(w \in B(x,r)\) and \(w \in B(y,s)\), so \(B(z,t) \subset U\).

Thus for every point \(z \in U\), an open ball around \(z\) is included in \(U\), and \(U\) is the union of all open balls which it includes.

Now we show that \(\mathcal T\) is a topology, where \(\mathcal T\) be the collection of all unions of open balls.

By definition, \(U \in \mathcal T\). Suppose \(V, W \in \mathcal T\) so \(V = \bigcup \mathcal A\) and \(W = \bigcup \mathcal B\) where \(\mathcal A, B\) are collections of open balls. Then

\[ V \cap W = \bigcup \{A \cap B: A \in \mathcal A, B \in \mathcal B\} \]

Thus \(V \cap W \in \mathcal T\). The empty set is in \(\mathcal T\) (as an empty union), and \(X\) is the union of all balls. Clearly, any union of sets in \(\mathcal T\) is in \(\mathcal T\). Thus \(\mathcal T\) is a topology. Also clearly, the balls form a base for it.

We prove the latter by showing that any neighborhood of \(x\) contains at least one of these balls.

Suppose \(x \in U \in \mathcal T\). Then for some \(y\) and \(r > 0, x \in B(y,r) \subset U\). Let \(s := r−d(x,y)\). Then \(s > 0\) and \(B(x,s) ⊂ U\), so the set of all balls with center at \(x\) is a neighborhood-base at \(x\).

Definition 1.5 The topology \(\mathcal T\) having the collection defined in Theorem 1.1 as a base is called a (pseudo)metric topology. If \(d\) is a metric, then \(\mathcal T\) is said to be metrizable and to be metrized by \(d\).

Example 1.4 On \(\mathbb R\), the topology metrized by the usual metric \(d(x,y) := |x - y|\) is the usual topology on \(\mathbb R\); namely, the topology with a base given by all open intervals \((a, b)\).