Are distinctions in definitions of "finite" material in, eg, topology or measure theory? There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set:
(Richard Dedekind) Every one-to-one function from S into itself is onto.
(Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion.
And there are more.
Do these distinctions matter when considering definitions of fundamental categories like topological spaces or measure spaces? For example, 
In topology: Why do we require a topological space to be closed under finite intersection?
In measure theory: "A measure is continuous from above if [given measurable sets and closed under intersection] at least one set has finite measure," alternatively, in the definition of sigma-finite measures (same article). 
Do the various definitions of finiteness lead to non-isomorphic or non-equivalent categories? 
My question is not specifically about topology or measure theory, but these are basic definitions introduced at undergrad level, so I thought, better to understand the context via basic examples.
 A: The various non-equivalent notions of finiteness you allude to refer to definitions of 'finite set'. The concepts of finiteness you mention that are relevant to analysis (in general) refer to a number being finite. It is common-place in analysis to assume the standard models for Peano Arithmetic and the real numbers. Both of these are categorical models (that means that any two models are isomorphic). Then, all the elements in a model of PA are finite and all the elements in $\mathbb {R}$ are finite as well. It is common to consider the extended real numbers $\mathbb {R} \cup \{\pm \infty\}$. Then $\pm \infty$ are by convention infinite numbers. 
So, there is little (if any) relation between the different definitions of 'finite set' and the notion if measure theory of a set having finite measure or in topology for taking finite intersections. Finite in the former means a real number (and not $\pm \infty$) while in the latter it means 'perform intersection a finite number of times, counted using some natural number'. 
