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Concept
List of topologies
Summary
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.
Discrete topology − All subsets are open.
Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open.
Cardinality and Ordinal number
Cocountable topology
Given a topological space the on is the topology having as a subbasis the union of τ and the family of all subsets of whose complements in are countable.
Cofinite topology
Double-pointed cofinite topology
Ordinal number topology
Pseudo-arc
Ran space
Tychonoff plank
Discrete two-point space − The simplest example of a totally disconnected discrete space.
Either–or topology
Finite topological space
Pseudocircle − A finite topological space on 4 elements that fails to satisfy any separation axiom besides T0. However, from the viewpoint of algebraic topology, it has the remarkable property that it is indistinguishable from the circle
Sierpiński space, also called the connected two-point set − A 2-point set with the particular point topology
Arens–Fort space − A Hausdorff, regular, normal space that is not first-countable or compact. It has an element (i.e. ) for which there is no sequence in that converges to but there is a sequence in such that is a cluster point of
Arithmetic progression topologies
The Baire space − with the product topology, where denotes the natural numbers endowed with the discrete topology. It is the space of all sequences of natural numbers.
Divisor topology
Partition topology
Deleted integer topology
Odd–even topology
List of fractals by Hausdorff dimension and Fractal
Apollonian gasket
Cantor set − A subset of the closed interval with remarkable properties.
Official source
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Finite topological space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions". Let be a finite set.
Comb space
In mathematics, particularly topology, a comb space is a particular subspace of that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation on the comb space. Consider with its standard topology and let K be the set . The set C defined by: considered as a subspace of equipped with the subspace topology is known as the comb space.
Extension topology
In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described in the sections below. Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose open sets are of the form A ∪ Q, where A is an open set of X and Q is a subset of P. The closed sets of X ∪ P are of the form B ∪ Q, where B is a closed set of X and Q is a subset of P.