Uniform boundedness of functions in non-metrizable spaces How can we define uniform boundedness of a function in a non-metrizable space?

For metric spaces, from Wikipedia (Uniform boundedness):

Let $Y$ be a metric space with metric $d$, then the set
$$\mathcal F=\{f_i: X \to Y, i\in I\}$$
is called uniformly bounded if there exists an element $a$ from $Y$ and a real number $M$ such that
$$d(f_i(x), a) \leq M \qquad \forall i \in I \quad \forall x \in X $$

 A: The usual notion for a space, where boundedness can be discussed is a bornological space, where we are given a family of sets, which we consider "small" or "bounded" in some sence. Recall

Definition 1. Let $X$ be a set, a family $\beta \subseteq \def\P{\mathfrak P}\P(X)$ is called bornology on $X$, if the following holds
  (i) $\bigcup \beta = X$ (every point lies in a bounded set)
  (ii) If $A \subseteq B$ and $B \in \beta$, then $A \in \beta$ (subsets of bounded sets are bounded).
  (iii) If $B_1, B_2 \in \beta$, then $B_1 \cup B_2 \in \beta$ (finite unions of bounded sets are bounded).
  Elements of $\beta$ are called ($\beta$-)bounded sets, the pair $(X,\beta)$ a bornological space.
Definition 2. Let $(X,\beta_X)$, $(Y, \beta_Y)$ be bornological spaces, a map $f \colon X \to Y$ is called bounded (or bornological, as that is not your version of bounded), if $f[\beta_X]\subseteq \beta_Y$, that is if the image of bounded sets is bounded.

Examples include the following:


*

*For every set $X$, $\P(X)$ and $\mathfrak F(X) := \{A \in \P(X): A \text{ is finite}\}$ are bornologies on $X$.

*For a metric space $(X,d)$, the $d$-bounded sets in the classical sence 
$$ \beta_d := \{A \in \P(X)\mid \exists a\in X,M: A \subseteq B_M(a)\} $$
form a bornology.

*To give a more interesting example, we consider a topological vector space $X$. A set $B \subseteq X$ is called (von Neumann-)bounded, if it is absorbed by all neighbourhoods of zero, that is: 
$$ \forall U \subseteq X: 0 \in \operatorname{int}U \implies \exists \lambda > 0 : B \subseteq \lambda U $$
The in these sense bounded sets, form a bornology on $X$, and we have (as for normed spaces:


Theorem. If $T \colon X \to Y$ is a continuous linear map between topological vector spaces, then it is bounded.
In this realm, you can of course also discuss the uniform version of being bounded, just translate the metric version you have above:

Definition. Let $(Y, \beta)$ be a bornological space, $X$ a set. A family of maps $f_i \colon X \to Y$ is called uniformly bounded, if there is a $B \in \beta$ such that $f_i[X]\subseteq B$, for all $i \in I$.

If you want to use that for a non-metric topological space (not that for metrizable spaces, the bornology does not depend only on the topology, but on the choice of the metric) $Y$, you have to say/define your bornology, that is to define what a bounded set in $Y$ is, examples are: $\P(X)$ (not very interesting so), $\mathfrak F(X)$ (makes a map bounded only if its image is finite) and (which in some sense gives Tien's example) and is a bornology if $Y$ is $T_1$: 
$$ \beta := \{ U\subseteq Y \mid \bar U \text{ is compact.}\} $$
