Introduction to Topology

1.1 What is Topology?

Topology is the branch of mathematics that studies properties of spaces that are preserved under continuous deformations. Two objects are considered topologically equivalent if one can be continuously deformed into the other without tearing or gluing.

The classic informal example: a coffee mug is topologically equivalent to a donut (torus). Both have exactly one hole, and one can be smoothly deformed into the other.

Formally, topology generalises the notions of closeness, continuity, and convergence from analysis and geometry, stripping away the rigid structure of distance and angles.

1.2 Motivation

Topology arises by definition from several directions:

• Analysis: The $\varepsilon$-$\delta$ definition of continuity in $\mathbb{R}^n$ depends only on open sets, not on the specific metric. Topology abstracts this to general spaces.
• Geometry: Many geometric properties (e.g., the number of holes in a surface) are invariant under continuous deformations.
• Algebra: Topological spaces carry algebraic invariants (fundamental group, homology groups) that classify spaces up to topological equivalence.

1.3 Historical Remarks

Key figures include Euler (Königsberg bridges, 1736), Riemann (Riemann surfaces, 1850s), Cantor (set theory, 1870s–80s), Poincaré (algebraic topology, 1890s), Hausdorff (metric and topological spaces, 1914), and Brouwer (fixed point theorem, 1911).