At it is well know, the von Neumann universe (or von Neumann hierarchy of sets) $V$ is a class of hereditary well-founded sets which can be used as a model for Zermelo–Fraenkel set theory (ZFC). This was stuff from the 1930s. Then in the 1960s, Paul Cohen invented the technique of forcing, and constructing a forcing extension $V[G]$ of $V$ using some Countable Chanin Condition forcing $P$. Then in the new universe $V[G]$, we have exotic new sets satisfying the exioms of ZFC.
In particular, in $V[G]$ we can "force" the existence of a set of reals $\mathbb{R}$ which does not satisfy the Continuum Hypothesis. In the original construction of Cohen, back to 1966, the set $\mathbb{R}$ in $V[G]$ has cardinal $\aleph_2$ (instead the usual assumption that $\text{card}(\mathbb{R}) = \aleph_1$). Then in the "forced universe" $V[G]$, the continuum hypothesis no longer holds, because between the rationals $\mathbb{Q}$, which have cardinal $\aleph_0$ and the whole set of reals, which now has cardinal $\aleph_2$ is possible to find sets $A$, with $\text{card}(\mathbb{Q}) < \text{card}(A) < \text{card}(\mathbb{R})$.
I wonder if someone have developed any application of Cohen reals, for which $\text{card}(\mathbb{R}) = \aleph_2$, other than model theory, or the foundations of set theory. For example, what about integration, measeure theory, real analysis or topology?