I am looking for the meaning of topological an algebraic properties of the space of continuous function $C(X)$ where $C(X)$ is a normed vector space with the supremum norm, in general terms (explanation for a non expert in the topic) and a few examples if possible.
I used google and the results were books that you need to buy to see them.
I found also Topological property but I guess that is a different concept since it is defined inside a topological space.
An example of an algebraic property is that $C(X)$ is again a vector space, since finite linear combinations of continuous functions are continuous.
Topological properties are numerous: you might want to search for general or point set topology (there are many free lecture notes online).
For example, since $\Bbb R$ is complete, so is $C(X)$ with the sup norm. (I’m assuming $C(X)$ is the space of real valued continuous functions $f : X \to \Bbb R$, but the same applies if the field is $\Bbb C$.)
There are more examples here: https://en.m.wikipedia.org/wiki/Continuous_functions_on_a_compact_Hausdorff_space