Editing General topology (section)

==Main areas of research==
[[Image:Peanocurve.svg|400px|thumb|Three iterations of a Peano curve construction, whose limit is a space-filling curve. The Peano curve is studied in [[continuum theory]], a branch of '''general topology'''.]]

===Continuum theory===
{{Main|Continuum (topology)}}
A '''continuum''' (pl ''continua'') is a nonempty [[compact space|compact]] [[connected space|connected]] [[metric space]], or less frequently, a [[compact space|compact]] [[connected space|connected]] [[Hausdorff space]]. '''Continuum theory''' is the branch of topology devoted to the study of continua.  These objects arise frequently in nearly all areas of topology and [[mathematical analysis|analysis]], and their properties are strong enough to yield many 'geometric' features.

===Dynamical systems===
{{Main|Topological dynamics}}
Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change.  Many examples with applications to physics and other areas of math include [[fluid dynamics]], [[dynamical billiards|billiards]] and [[geometric flow|flows]] on manifolds.  The topological characteristics of [[fractal]]s in fractal geometry, of [[Julia set]]s and the [[Mandelbrot set]] arising in [[complex dynamics]], and of [[attractor]]s in differential equations are often critical to understanding these systems.{{Citation needed|date=December 2019}}

===Pointless topology===
{{Main|Pointless topology}}
'''Pointless topology''' (also called '''point-free''' or '''pointfree topology''') is an approach to [[topology]] that avoids mentioning points. The name 'pointless topology' is due to [[John von Neumann]].<ref>Garrett Birkhoff, ''VON NEUMANN AND LATTICE THEORY'', ''John Von Neumann 1903-1957'', J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5</ref>  The ideas of pointless topology are closely related to [[mereotopology|mereotopologies]], in which regions (sets) are treated as foundational without explicit reference to underlying point sets.

===Dimension theory===
{{Main|Dimension theory}}
'''Dimension theory''' is a branch of general topology dealing with [[dimensional invariant]]s of [[topological space]]s.

===Topological algebras===
{{Main|Topological algebra}}
A '''topological algebra''' ''A'' over a [[topological field]] '''K''' is a [[topological vector space]] together with a continuous multiplication

:<math>\cdot :A\times A \longrightarrow A</math>
:<math>(a,b)\longmapsto a\cdot b</math>

that makes it an [[algebra over a field|algebra]] over '''K'''. A unital [[associative algebra|associative]] topological algebra is a [[topological ring]].

The term was coined by [[David van Dantzig]]; it appears in the title of his [[Thesis|doctoral dissertation]] (1931).

===Metrizability theory===
{{Main|Metrization theorem}}
In [[topology]] and related areas of [[mathematics]], a '''metrizable space''' is a [[topological space]] that is [[homeomorphism|homeomorphic]] to a [[metric space]]. That is, a topological space <math>(X,\tau)</math> is said to be metrizable if there is a metric 
:<math>d\colon X \times X \to [0,\infty)</math>

such that the topology induced by ''d'' is <math>\tau</math>.  '''Metrization theorems''' are [[theorem]]s that give [[sufficient condition]]s for a topological space to be metrizable.

===Set-theoretic topology===
{{Main|Set-theoretic topology}}
Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of [[Zermelo–Fraenkel set theory]] (ZFC). A famous problem is [[Moore space (topology)#Normal Moore space conjecture|the normal Moore space question]], a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.