nLab examples in topology

Basic examples

• sphere, ball, torus, real line, real n-dimensional space, interval, long line

Many classes come from standard constructions from other spaces or maps like

• quotient space, subspace (as subset with induced topology), loop space, free loop space, suspension, mapping cylinder, mapping cone, quotient space, Fadell's configuration space

or from data which includes some parameters or algebraic data as input like

• Grassmanian?s, flag varieties

and as representing object of some functor, or (more or less the same), as moduli spaces of some structures

• classifying space, Milnor construction

There are also classes of spaces which are often presented by standard recipes. For examples, manifolds may be presented by manifold atlases

Locally badly behaved spaces include

• Sierpinski space, Warsaw circle, fractal, solenoid

For a compilation of examples important in algebraic topology see

• Neil Strickland, A bestiary of topological objects (pdf)

Many examples are mentioned, or searched for, in the compendium of open problems

• J. van Mill, George M. Reed (editors), Open problems in topology, North Holland 1990, pdf