The proverbial motivating example is the surface of a torus and that of a mug: these are isomorphic as topological spaces, because, abstractly, the surface of the mug can be deformed continuously to become the standard torus: the continuous cohesion among the collections of points of the two surfaces is the same.
There is a slight generalization of the notion of topological space to that of a locale, which consists of dropping the assumption that all neighbourhoods are explicitly or even necessarily supported by points. In this form the definition is quite fundamental and can be naturally motivated from just pure logic – as the formal dual of frames – as well as, and dually, from category theory in its variant as topos theory – by the notion of (0,1)-toposes.
Topological spaces equipped with extra property and structure form the fundament of much of geometry. For instance a topological space locally isomorphic to a Cartesian space is a manifold. A topological space equipped with a notion of smooth functions into it is a diffeological space. The intersection of these two notions is that of a smooth manifold on which differential geometry is based. And so on.
We present first the
and then a list of different
Finally we mention genuine
The word ‘topology’ sometimes means the study of topological spaces but here it means the collection of open sets in a topological space. In particular, if someone says ‘Let be a topology on ’, then they mean ‘Let be equipped with the structure of a topological space, and let be the collection of open sets in this space’.
Since itself is the intersection of zero subsets, it is open, and since the empty set is the union of zero subsets, it is also open. Moreover, every open subset of contains the empty set and is contained in
Since by definition the elements in this poset are closed under finite meets (intersection) and arbitrary joins (unions), this poset of open subsets defining a topology is a frame, the frame of opens of .
There are many equivalent ways to define a topological space. A non-exhaustive list follows:
A set with a frame of open sets (as above).
A set with a co-frame of closed sets (the complements of the open sets) satisfying dual axioms.
A set with any collection of subsets whatsoever, to be thought of as a subbase for a topology.
A relational β-module; that is, a lax algebra? of the monad of ultrafilters on the (1,2)-category Rel of sets and binary relations. More explicitly, this means a set together with a relation called “convergence” between ultrafilters and points satisfying certain axioms. This exhibits it as a special sort of generalized multicategory, and also as a special sort of pseudotopological space.
The definition of topological space was a matter of some debate, especially about 100 years ago. Our definition is due to Bourbaki, so may be called Bourbaki spaces.
For some purposes, including homotopy theory, it is important to use nice topological spaces (such as sequential topological spaces) and/or a nice category of spaces (such as compactly generated spaces), or indeed to directly use a model of -groupoids (such as simplicial sets).
See at topology.