The Weak Topology Induced by F

Let $X$ be any nonempty set and let $\mathcal F$ be a collection of real-valued functions defined on $X$. For each $\epsilon > 0$, each finite subcollection $F \subseteq \mathcal F$, and each $x \in X$ let:

(1)

\begin{align} \quad N_{\epsilon, F} (x) = \{ y \in X : |f(x) - f(y)| < \epsilon, \forall f \in F \} \end{align}

Then for each $x \in X$, the following collection of sets will form a base at $x$:

(2)

\begin{align} \quad \{ N_{\epsilon, F}(x) : \epsilon > 0, F \subseteq \mathcal F \: \mathrm{finite} \} \end{align}

The topology on $X$ that is generated by these local bases has a special name.

Proposition 1: Let $X$ be a nonempty set and let $\mathcal F$ be a collection of functions on $X$. For each $x \in X$, the collection $\{ N_{\epsilon, F}(x) : \epsilon > 0, x \in X, F \subseteq \mathcal F \: \mathrm{finite} \}$ is a base for some topology $\tau$ on $X$.

Recall from the Bases for a Topology page that a collection $\mathcal B$ of subsets of a set $X$ is a base for some topology $\tau$ on $X$ if and only if $X$ is covered by $\mathcal B$ and for all $B_1, B_2 \in \mathcal B$ and $x \in B_1 \cap B_2$ where exists a $B \in \mathcal B$ such that $x \in B \subseteq B_1 \cap B_2$. We use this result in proving the proposition above.

• Proof: Let $f \in \mathcal F$ and let $\epsilon = 1 > 0$. Then $x \in N_{1, \{ f \}}(x)$ for each $x \in X$ and so $X = \bigcup_{x \in X} N_{1, \{ f \}}(x)$ which shows that $X$ is covered by $\{ N_{\epsilon, F}(x) : \epsilon > 0, x \in X, F \subseteq \mathcal F \: \mathrm{finite} \}$.

• Secondly let $\epsilon_1, \epsilon_2 > 0$, $F_1, F_2 \subseteq \mathcal F$ be finite, and let $x_1, x_2 \in X$. Suppose that:

(3)

\begin{align} \quad x \in N_{\epsilon_1, F_1}(x_1) \cap N_{\epsilon_2, F_2}(x_2) \end{align}

Then by definition $x \in N_{\epsilon_1, F_1}(x_1)$ and $x \in N_{\epsilon_2, F_2}(x_2)$ and so:

(4)

\begin{align} \quad |f(x) - f(x_1)| < \epsilon_1, \quad \forall f \in F_1 & \quad (*) \\ \quad |g(x) - g(x_2)| < \epsilon_2, \quad \forall g \in F_2 & \quad (**) \end{align}

Let $\delta > 0$ be such that:

(5)

\begin{align} \quad 0 < \delta < \min_{f \in F_1, g \in F_2} \{ \epsilon_1 - |f(x) - f(x_1)|, \epsilon_2 - |g(x) - g(x_2)| \} \end{align}

Now we have that $\delta > 0$ and $F_1 \cup F_2$ is finite (since $F_1$ and $F_2$ are both finite sets). So if $x' \in N_{\delta, F_1 \cup F_2}(x)$ we have that for all $f \in F_1$:

(6)

\begin{align} \quad | f(x') - f(x_1) | &\leq | f(x') - f(x) | + | f(x) - f(x_1) | \\ & < \delta + |f(x) - f(x_1)| \\ & < (\epsilon_1 - |f(x) - f(x_1)|) + |f(x) - f(x_1)| \\ & < \epsilon_1 \end{align}

And we have that for all $g \in F_2$ that:

(7)

\begin{align} \quad | g(x') - g(x_2) | & \leq | g(x') - g(x)| + |g(x) - g(x_2)| \\ & < \delta + |g(x) - g(x_2)| \\ & < (\epsilon_2 - |g(x) - g(x_2)|) + |g(x) - g(x_2)| \\ & < \epsilon_2 \end{align}

Therefore $x' \in N_{\epsilon_1, F_1}(x_1) \cap N_{\epsilon_2, F_2}(x_2)$. Thus:

(8)

\begin{align} \quad N_{\delta, F_1 \cup F_2} (x) \subseteq N_{\epsilon_1, F_1}(x_1) \cap N_{\epsilon_2, F_2}(x_2) \end{align}

So indeed, $\mathcal B$ is a base for some topology $\tau$ on $X$. $\blacksquare$

Proposition 1 tells us that the collection of sets $\{ N_{\epsilon, F}(x) : \epsilon > 0, x \in X, F \subseteq \mathcal F \: \mathrm{finite} \}$ generates a topology on $X$. This topology will be given a special name which we define below.

Definition: Let $X$ be a nonempty set and let $\mathcal F$ be a collection of real-valued functions on $X$. The Weak Topology Induced by $\mathcal F$ or the $\mathcal F$-Weak Topology on $X$ is the topology generated by the local bases for each $x \in X$ given by $\{ N_{\epsilon, F}(x) : \epsilon > 0, F \subseteq \mathcal F \: \mathrm{finite} \}$. It is the weakest topology (the initial topology) which makes every function in $\mathcal F$ continuous with respect to the topology.

Local Base for x

For each $x \in X$, a local base for $x$ in the $\mathcal F$-weak topology consists of sets of the form:

(9)

\begin{align} \quad N_{\epsilon, \{ f_1, f_2, ..., f_n \}}(x) = \{ y \in X : |f_k(x) - f_k(y)| < \epsilon, \: 1 \leq k \leq n \} \end{align}

F-Weak Convergence

It is useful to characterize what it means for a sequence of points $(x_n)$ to converge to a point $x$ with respect to the $\mathcal F$-Weak topology on $X$. We define this below in the following proposition.

Proposition 2: Let $X$ be equipped with the $\mathcal F$-weak topology. A sequence of points $(x_n)$ $\mathcal F$-weakly converges to $x \in X$ if $\displaystyle{\lim_{n \to \infty} f(x_n) = f(x)}$ for every $f \in \mathcal F$.

Recall that if $(X, \tau)$ is a topological space then a sequence of points $(x_n)$ converges to $x \in X$ if for every $U \in \tau$ with $x \in U$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then $x_n \in U$.