List of concrete topologies and topological spaces
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
Counter-examples (general topology)
The following topologies are a known source of counterexamples for point-set topology.
- Alexandroff plank
- Appert topology − A Hausdorff, perfectly normal (T6), zero-dimensional space that is countable, but neither first countable, locally compact, nor countably compact.
- Arens square
- Bullet-riddled square - The space where is the set of bullets. Neither of these sets is Jordan measurable although both are Lebesgue measurable.
- Cantor tree
- Comb space
- Dieudonné plank
- Double origin topology
- Dunce hat (topology)
- Either–or topology
- Excluded point topology − A topological space where the open sets are defined in terms of the exclusion of a particular point.
- Fort space
- Half-disk topology
- Hilbert cube − with the product topology.
- Infinite broom
- Integer broom topology
- K-topology
- Knaster–Kuratowski fan
- Long line (topology)
- Moore plane, also called the Niemytzki plane − A first countable, separable, completely regular, Hausdorff, Moore space that is not normal, Lindelöf, metrizable, second countable, nor locally compact. It also an uncountable closed subspace with the discrete topology.
- Nested interval topology
- Overlapping interval topology − Second countable space that is T0 but not T1.
- Particular point topology − Assuming the set is infinite, then contains a non-closed compact subset whose closure is not compact and moreover, it is neither metacompact nor paracompact.
- Rational sequence topology
- Sorgenfrey line, which is endowed with lower limit topology − It is Hausdorff, perfectly normal, first-countable, separable, paracompact, Lindelöf, Baire, and a Moore space but not metrizable, second-countable, σ-compact, nor locally compact.
- Sorgenfrey plane, which is the product of two copies of the Sorgenfrey line − A Moore space that is neither normal, paracompact, nor second countable.
- Topologist's sine curve
- Tychonoff plank
- Vague topology
- Warsaw circle