Topological Spaces (𝑋,𝜏)
- is a type of mathematical space
- is a tuple (𝑋,𝜏) where:
Topological Space - Definitions
Definition via Open Sets (Topology (𝜏))
see Topology (𝜏)
Definition via Closed Sets
Definition
Using de Morgan’s laws, the above axioms defining open sets become axioms defining closed sets:
- The empty set and 𝑋 are closed.
- The intersection of any collection of closed sets is also closed.
- The union of any finite number of closed sets is also closed.
Using these axioms, another way to define a topological space is as a set 𝑋 together with a collection 𝜏 of closed subsets of 𝑋. Thus the sets in the topology 𝜏 are the closed sets, and their complements in 𝑋 are the open sets.
Definition via Neighborhoods
Definition
A topological space is a pair (𝑋,𝒩) where:
- 𝑋 be a set
- 𝒩 be a function assigning to each 𝑥∊𝑋 a non-empty collection 𝒩(𝑥) of subsets of 𝑋. The elements of 𝒩(𝑥) will be called neighbourhoods of 𝑥 with respect to 𝒩 (or, simply, neighbourhoods of 𝑥). The function 𝒩 is called a neighbourhood topology if the axioms below are satisfied:
- If 𝑁 is a neighbourhood of 𝑥 (i.e., 𝑁∈𝒩(𝑥)), then 𝑥∈𝑁. In other words, each point belongs to every one of its neighbourhoods.
- If 𝑁 is a subset of 𝑋 and includes a neighbourhood of 𝑥, then 𝑁 is a neighbourhood of 𝑥 (i.e. every superset of a neighbourhood of a point 𝑥∈𝑋 is again a neighbourhood of 𝑥)
- The intersection of two neighbourhoods of 𝑥 is a neighbourhood of 𝑥
- Any neighbourhood 𝑁 of 𝑥 includes a neighbourhood 𝑀 of 𝑥 such that 𝑁 is a neighbourhood of each point of 𝑀
The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of 𝑋.
Examples
A standard example of such a system of neighbourhoods is for the real line ℝ, where a subset 𝑁 of ℝ is defined to be a neighbourhood of a real number 𝑥 if it includes an open interval containing 𝑥.
Given such a structure, a subset 𝑈 of 𝑋 is defined to be open if 𝑈 is a neighbourhood of all points in 𝑈. The open sets then satisfy the axioms given below. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining 𝑁 to be a neighbourhood of 𝑥 if 𝑁 includes an open set 𝑈 such that 𝑥∈𝑈.
Topological Spaces - Types
- All Metric Spaces (𝑋,𝑑) can induce a topology (𝜏) defined as distance metric topology (𝜏𝑑)
- Topological Vector Spaces (TVS)
- Indiscrete Topological Spaces
- Discrete Topological Spaces
- Function Spaces
- Proximity Spaces
- Uniform Spaces
- Cauchy Spaces
- Convergence Spaces
Subpages
- Algebraic Topology
- Kolmogorov Classification
- Topological Spaces (Boundary - Boundaryless)
- Topological Spaces (Compactness／Compact - Sequential Compactness - Sequentially Compact - Limit Point Compactness)
- Topological Spaces (Cover／Covering - Open Cover／Covering - Subcover／Subcovering)
- Topological Spaces (Limit Points - Accumulation Points - Cluster Points)
- Topological Spaces (Neighborhoods of a Point／Set)
- Topological Spaces (Open Sets - Closed Sets - Clopen Sets - Closures)
- Topology
- Transformations Between Topological Spaces