  We study asymptotic statistical inference in the space of bounded functions
endowed with the supremums norm over an arbitrary metric space $S$ using a
novel concept: Simultaneous COnfidence Region of Excursion (SCoRE) Sets. They
simultaneously quantify the uncertainty of several lower and upper excursion
sets of a target function. We investigate their connection to multiple
hypothesis tests controlling the familywise error rate in the strong sense and
show that they grant a unifying perspective on several statistical inference
tools such as simultaneous confidence bands, quantification of uncertainties in
level set estimation, for example, CoPE sets, and multiple hypothesis testing
over $S$, for example, finding relevant differences or regions of equivalence
within $S$. In particular, our abstract setting allows us to refine and reduce
the assumptions in recent articles on CoPE sets and relevance and equivalence
testing using the supremums norm.
