Topology Induced by $b_s$-pseudo metric

Question

1. Given $s\geq1$, $X$ is nonempty set and $d: X\times X \rightarrow [0,\infty)$ is $b_s$-pseudo metric on $X$ if for $x,y,z\in X$

i. $d(x,x)=0$

ii. $d(x,y)=d(y,x)$

iii. $d(x,y)\leq s[d(x,z)+d(z,y)]$

2. $F=\{d_n:n\in\mathbb{N}\}$ is a family of $b_s$-pseudo metric. $X$ is non-empty set.

The pair $(X,\tau(X))$ is called a $b_s$-gauge space where $\tau(X)$ is the topology having subbases in the family $$B(F)=\{B(x,d_n,\varepsilon);x\in X, d_n\in F,\ and\ \varepsilon>0\}$$

I don't understand the 'topology having subbases in ...' part. What is the subbases definiton here? How do we induced a topology from subbases?

If I have $C([0,\infty),\mathbb{R})$ is a set of all continuous function and the family of $b_2$-pseudo metric $F=\{d_k(x,y):k\in \mathbb{N}\}$ where $$d_k(x,y) = \max_{t\in [0,k]} (x(t)-y(t))^2$$ for every $x,y\in X$ and $k\in\mathbb{N}$, what is the example of the topology induced by $F$?

Answer 1

What it really means is that $B(F)$ is a subbase for $\tau$. $B(F)$ is simply the set of all open $\epsilon$-balls for all of the $b_s$-pseudometrics in $F$.

The topology induced by $B(F)$ is formed in the standard way. First let

$$\mathscr{B}=\left\{\bigcap\mathscr{F}:\mathscr{F}\subseteq B(F)\text{ is finite}\right\}\,,$$

the set of all intersections of finitely many of the open balls in $B(F)$; then $\mathscr{B}$ is a base for $\tau$, i.e., $\tau$ is the set of all unions of members of $\mathscr{B}$.

As for your example, let $x\in C([0,\infty),\Bbb R)$, and let $A\subseteq\Bbb N$ be finite and non-empty. For each $k\in A$ let $\epsilon_k>0$. Then

$$\left\{y\in C([0,\infty),\Bbb R):d_k(x,y)<\epsilon_k\text{ for all }k\in A\right\}\tag{1}$$

is the intersection of finitely many members of $B(F)$ and is therefore a basic open nbhd of $x$. Let $\epsilon=\min\{\epsilon_k:k\in A\}$ and $m=\max A$; then

$$\begin{align*} B(x,d_m,\epsilon)&=\{y\in C([0,\infty),\Bbb R):d_m(x,y)<\epsilon\}\\ &=\left\{y\in C([0,\infty),\Bbb R):\max_{t\in[0,m]}\big(x(t)-y(t)\big)^2<\epsilon\right\} \end{align*}$$

is an open nbhd of $x$ contained in the set in $(1)$, so in this case the subbase $B(F)$ is actually a base for the topology.