What does Conway mean by $C_b(X)$ is 'very small'

Question

In Conway's A Course in Functional Analysis, $2$nd edition, page $137,$ section 'An Application: The Stone-Cech Compactification,' the author began the section with the following paragraph.

Let $X$ be a topological space and considering the Banach space $C_b(X).$ Unless some assumptions is made regarding $X,$ it may be that $C_b(X)$ is 'very small.' If, for example, it is assumed that $X$ is completely regular, then $C_b(X)$ has many elements.

So my question is, what does the author mean by 'very small'? Also, what $X$ will satisfy such 'smallness'?

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Notation: For any topological space $X,$ let $C_b(X)$ be the space of all real-valued bounded contnouus functions on $X.$

Answer 1

$X$ can have very few continuous functions, e.g. if $X$ is indiscrete, so has the topology $\{\emptyset, X\}$ and $X$ has at least two points, then every continuous $f: X \to Y$ is constant whenever $Y$ is $T_0$ (certainly true for $Y=\mathbb{R}$): suppose $f$ were not constant and $f(x_1) = p_1 \neq p_2 = f(x_2)$, then in $Y$ we can find an open set $O$ such that $p_1 \in O$, $p_2 \notin O$ (or vice versa). But then $f^{-1}[O]$ is open in $X$, non-empty and not equal to $X$, which cannot be (by the definition of the indiscrete topology). So $f$ must be constant.

In fact there are even examples of $T_3$ (in particular Hausdorff) spaces $X$ such that all continuous functions from $X$ to $\mathbb{R}$ are constant. Note that constant functions to $\mathbb{R}$ are always continuous, so having only the constant ones is the bare minimum. Also then $C_b(X) \simeq \mathbb{R}$ both in such an example's case and the indiscrete space (and the case $|X|= 1$). So $C_b(X)$ does not give a lot of extra info on $X$, as it were, in these trivial cases. In fact for spaces that are completely regular/Tychonoff we have enough continuous bounded continuous functions on $X$ to avoid these trivialities and then it turns out that $C_b(X)$ can carry a lot of information on $X$, and this is exactly the condition Conway imposes. All metric spaces, or ordered spaces, or $T_0$ topological groups are completely regular, so almost all spaces one encounters "in practice" are OK, and have a large $C_b(X)$.