Chaotic and Discrete Topologies

THE TWO ENDS OF WHAT OPEN CAN MEAN

Every set \(M\) carries a smallest and a largest topology. The chaotic one calls only \(\emptyset\) and \(M\) open; the discrete one calls every subset open. Every other topology on \(M\) contains the first and is contained in the second.

chaotic only two subsets are open discrete every subset is open the eight subsets of M = {1, 2, 3}, twice over a filled dot means that element is in
The same eight subsets under the two extreme topologies.

The definitions

Definitions · chaotic and discrete topologies

The chaotic topology on \(M\), elsewhere called the indiscrete or trivial topology, is \(\mathcal{O}_{\text{chaotic}} := \{\emptyset, M\}\), the smallest collection the conditions on a topology permit.

The discrete topology on \(M\) is \(\mathcal{O}_{\text{discrete}} := \mathcal{P}(M)\), in which every subset is open.

Both really are topologies

Each has to hold \(\emptyset\) and \(M\), and be closed under intersections of two members and unions of any family of them. The chaotic one holds exactly those two sets, and every intersection or union of them is again one of them. The discrete one holds every subset, so every union and intersection of its members is one of its members again. Neither check needs to know anything about \(M\).

What each one can tell apart

The chaotic topology cannot tell any two points apart: the only open set holding a point is the whole of \(M\), so no open set separates one point from another. The discrete topology tells every point from every other. With every subset open, every preimage is open, so every map out of \(M\) is continuous whatever the target and continuity says nothing.

On a set with fewer than two points the two coincide, and are then the only topology available.