tilajoukkoina

Tilajoukkoina is a Finnish mathematical term meaning “as topological spaces.” It is used when a mathematical object that can be endowed with a topology is treated in its role as a topological space rather than merely as a set or algebraic structure. The basic unit is tilajoukko (topological space), consisting of a set equipped with a topology—a collection of open subsets closed under arbitrary unions and finite intersections, including the empty set and the whole set.

The phrase tilajoukkoina appears in contexts where one wants to emphasize the topological structure. For example, k

katsotaan tilajoukkoina X ja Y means “we consider X and Y as topological spaces.” In such usage, statements and proofs may rely on the topology, not just the underlying set. This perspective is important in discussions of continuous maps, homeomorphisms, and categorical relationships between objects whose morphisms preserve topological structure.

When problems or theorems specify properties that depend on the topology, one tends to describe the objects tilajoukkoina to distinguish from considering only their underlying sets or other structures. For instance, a function f: X -> Y is continuous if it preserves open sets; discussing this property tilajoukkoina emphasizes the topological context.

See also: topologia (topology), tila (space), topologinen tila (topological space), alijoukko (subset), jatkuvuus (continuity), homeomorfia (homeomorphism).