Topològics

Topològics, sometimes referred to as topological spaces, are fundamental objects in topology, a branch of mathematics concerned with properties of space that are preserved under continuous deformations. A topological space is essentially a set equipped with a structure that allows for the definition of continuity. This structure is defined by a collection of subsets of the set, called open sets, which satisfy certain axioms. These axioms ensure that the notion of nearness or proximity between points can be consistently defined.

The axioms for open sets are: the empty set and the entire set must be open; the union of any number of open sets must be open; and the intersection of any finite number of open sets must be open. This collection of open sets forms the topology of the space. From open sets, other topological concepts like closed sets, neighborhoods, and continuity of functions can be derived. For instance, a function between two topological spaces is continuous if the preimage of every open set in the codomain is an open set in the domain.

Topologies can vary significantly. For a given set, one can define different topologies, ranging from the indiscrete topology (where only the empty set and the whole set are open) to the discrete topology (where every subset is open). The choice of topology greatly influences the topological properties of the space. For example, a set with the discrete topology is always Hausdorff, meaning distinct points have disjoint neighborhoods, whereas a set with the indiscrete topology is generally not Hausdorff.