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)Then for each $x \in X$, the following collection of sets will form a base at $x$:
(2)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:
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)Let $\delta > 0$ be such that:
(5)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)And we have that for all $g \in F_2$ that:
(7)Therefore $x' \in N_{\epsilon_1, F_1}(x_1) \cap N_{\epsilon_2, F_2}(x_2)$. Thus:
(8)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)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 $x_n \in U$.