An overview of the fundamental concepts in topology, including sets, topological spaces, open and closed sets, neighborhoods, and basic definitions.
Introduction to Topology – Study Notes
| Concept | Description | Example |
|---|---|---|
| Set | A collection of points (elements). | (X = {1,2,3}) |
| Topology on a set | A collection (\mathcal{T}) of subsets of (X) satisfying: 1. (\emptyset, X \in \mathcal{T}) 2. Arbitrary unions of members of (\mathcal{T}) are in (\mathcal{T}) 3. Finite intersections of members of (\mathcal{T}) are in (\mathcal{T}). |
Standard topology on (\mathbb{R}): all open intervals ((a,b)). |
| Open set | Any member of the topology (\mathcal{T}). | ((0,1)) in (\mathbb{R}). |
| Closed set | Complement of an open set. | ([0,1]) is closed in (\mathbb{R}). |
| Neighbourhood | Any set containing an open set around a point. | For (x=2) in (\mathbb{R}), the interval ((1.5,2.5)) is a neighbourhood. |
| Basis | A subcollection (\mathcal{B}) of (\mathcal{T}) such that every open set is a union of basis elements. | In (\mathbb{R}), the set of all open intervals with rational endpoints forms a countable basis. |
| Subspace topology | If (Y\subseteq X), the topology on (Y) is ({U\cap Y \mid U\in\mathcal{T}}). | The interval ([0,1]) inherits the usual topology from (\mathbb{R}). |
| Product topology | For spaces (X) and (Y), the topology on (X\times Y) generated by products of open sets (U\times V). | (\mathbb{R}^2) with the Euclidean topology is a product of two copies of (\mathbb{R}). |
| Quotient topology | Given a surjection (q:X\to Y), a set (U\subseteq Y) is open iff (q^{-1}(U)) is open in (X). | Collapsing the boundary of a disk to a point yields a sphere. |