Topology

References: Most of the definitions and notation in the section are based on ¹

A topological space is a set for which a valid topology has been defined: the topology determines which subsets of the topological space are open and closed. In this way the concept of open and closed subsets on the real number line (such as (0,1) and [1,2]) are generalized to arbitrary sets.

Formally, a topology on a set A is a collection 𝒯 of subsets of A fufiling the criteria:

1. The empty set and the entire set A are both in 𝒯.

2. The union of an arbitrary number of elements of 𝒯 is also in 𝒯.

3. The intersection of a finite number of elements of 𝒯 is also in 𝒯.

If a subset B of A is a member of 𝒯 then B is an open set under the topology 𝒯.

Coarseness and Fineness are ways of comparing two topologies on the same space. 𝒯′ is finer than 𝒯 if 𝒯 is a subset of 𝒯′ (and 𝒯 is coarser); it is strictly finer if it is a proper subset (and 𝒯 is strictly coarser). Two sets are comprable if either 𝒯 ∈ 𝒯′ or 𝒯′ ∈ 𝒯. Smaller and larger are somtimes used instead of finer and coarser.

Topologies can be generated from a basis.

TODO: Hausdorf

Frequently Used Topologies

Standard Topology

The standard topology on the real line is generated by the collection of all intervals (a,b) = {x|a < x < b} This is the usual definition for open sets on the real line.

Discrete Topology

The topology on a set A consisting of all points of A; in other words the power set of A.

Trivial/Indiscrete Topology

The topology on a set A consisting of only the empty set and A itself. Not super interesting but it’s always there (when A isn’t empty).

Finite Complement Topology (𝒯_f)

The topology on a set A consisting of the empty set any subset U such that A − U has a finite number of elements.

Lower Limit Topology (ℝ_𝓁)

The lower limit topology on the real line is generated by the collection of all half open intervals [a, b) = {x|a ≤ x < b} ℝ_𝓁 is strictly finer than the standard topology and is not comprable to ℝ_K.

K-Topology (ℝ_K)

Let K denote the set of all numbers 1/n where n is a positive integer. The K-topology on the real line is generated by the collection of all standard open intervals minus K. ℝ_K is strictly finer than the standard topology and is not comprable to ℝ_𝓁.

Order Topology

TODO