  In this paper we introduce a new kind of topological space, called
'structured space', which locally resembles various kinds of algebraic
structures. This can be useful, for instance, to locally study a space that
cannot be globally endowed with an algebraic structure using tools from
algebra. The definition of these spaces will be made more precise via one of
our main result, which involves the 'structure map'. This will also lead us to
a rigorous and unambiguous definition of algebraic structure. After showing
some examples which naturally arise in this context, we study various
properties and develop some theory for these new spaces; in particular, we
consider partitions (with respect to some measure $\mu$). We then prove one of
the most important Theorem of this paper (Theorem 4.1), which states that every
structured space, under some assumptions, induces a lattice, and conversely
every lattice induces a structured space satisfing such hypothesis. We conclude
with some relations with connected spaces.
