Convergence space (also called a ''limit space'' or ''Čech space'') is a generalization of topological spaces in which the notion of convergence of filters (or nets) is taken as primitive, rather than being derived from an open‑set structure. The concept provides a flexible framework for studying convergence phenomena that are not captured by ordinary topologies, and it plays an important role in general topology, functional analysis, and categorical topology.
Definition
A convergence space is a pair $(X,\to)$ where $X$ is a set and $\to$ is a relation between filters (or, equivalently, nets) on $X$ and points of $X$, written $\mathcal{F}\to x$, satisfying the following axioms for every filter $\mathcal{F},\mathcal{G}$ on $X$ and points $x,y\in X$:
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Reflexivity: The principal filter $\mathcal{U}_x:={A\subseteq X\mid x\in A}$ converges to $x$: $$ \mathcal{U}_x \to x . $$
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Isotony (Monotonicity): If $\mathcal{F}\subseteq\mathcal{G}$ and $\mathcal{F}\to x$, then $\mathcal{G}\to x$.
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Stability under Intersection: If $\mathcal{F}\to x$ and $\mathcal{G}\to x$, then the filter generated by $\mathcal{F}\cap\mathcal{G}$ also converges to $x$.
Equivalently, one may define a convergence space by specifying, for each point $x\in X$, a filter of neighborhoods $\mathcal{N}(x)$ (the set of all filters converging to $x$). The three axioms then translate into conditions that each $\mathcal{N}(x)$ be a filter on the set of all filters on $X$.
When the convergence relation is restricted to ultrafilters or nets, the same axioms give rise to the notions of ultrafilter convergence spaces and net convergence spaces, which are equivalent to the filter formulation.
Relation to Topological Spaces
Every topological space $(X,\tau)$ determines a convergence space by declaring that a filter $\mathcal{F}$ converges to $x$ iff $\mathcal{F}$ contains every neighbourhood of $x$: $$ \mathcal{F}\to x ;\Longleftrightarrow; \forall U\in\tau;(x\in U \implies U\in\mathcal{F}). $$ Conversely, a convergence space $(X,\to)$ is topologizable (i.e., arises from a topology) precisely when the following Kelley condition holds:
For all $x\in X$ and all subsets $A\subseteq X$, if every filter that converges to $x$ meets $A$, then $x$ belongs to the closure of $A$.
When the Kelley condition is satisfied, the family $$ \mathcal{O}_x:={U\subseteq X\mid \text{every filter converging to }x\text{ contains }U} $$ forms a neighbourhood filter, and the resulting topology $\tau$ satisfies $\mathcal{F}\to x$ iff $\mathcal{F}$ contains every $\tau$-neighbourhood of $x$.
Thus, the category Top of topological spaces embeds fully and faithfully into the category Conv of convergence spaces, but Conv is strictly larger.
Examples
| Example | Description |
|---|---|
| Topological convergence | Any topological space $(X,\tau)$ gives a convergence space as above. |
| Preorder convergence | Let $(P,\le)$ be a preordered set. Define $\mathcal{F}\to x$ iff every set in $\mathcal{F}$ contains a point $y\le x$. This yields a convergence space that need not be topologizable. |
| Convergence of sequences | On a set $X$, declare a filter generated by a sequence $(x_n)$ to converge to a limit $x$ if the usual sequence limit holds (e.g., in metric spaces). Extending this rule to all filters gives a convergence space that captures sequential convergence but may ignore non‑sequential limits. |
| Ultrafilter convergence | Define $\mathcal{U}\to x$ iff $x$ belongs to the limit of the ultrafilter $\mathcal{U}$ in a given compact Hausdorff space. This provides a model of Stone‑Čech compactification within convergence spaces. |
| Indiscrete convergence | The relation $\mathcal{F}\to x$ holds for every filter $\mathcal{F}$ and every point $x$. This is the maximal convergence structure, coarser than any topological one. |
| Discrete convergence | Only the principal filter at $x$ converges to $x$. This coincides with the usual discrete topology. |
Categorical Properties
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Category Conv – Objects are convergence spaces, morphisms are continuous maps: a function $f:X\to Y$ is continuous if whenever $\mathcal{F}\to x$ in $X$, the image filter $f[\mathcal{F}]$ converges to $f(x)$ in $Y$.
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Reflective subcategory – The full subcategory Top of topological spaces is reflective in Conv: the topologization functor $T: \text{Conv}\to\text{Top}$ sends a convergence space to the finest topology inducing the same convergence.
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Coreflective subcategory – The subcategory PreTop of pretopological spaces (where convergence is defined only for filters that contain a neighbourhood filter) is coreflective.
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Products and coproducts – Both limits and colimits exist in Conv and are constructed pointwise using filter operations; they extend the usual product and sum constructions in Top.
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Cartesian closed – Unlike Top, the category Conv is not cartesian closed in general; however, certain subcategories (e.g., of compactly generated convergence spaces) inherit a closed structure.
Applications
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Functional analysis – Convergence spaces provide a natural setting for studying weak and weak* convergence of functionals, especially when the underlying topological structure is inadequate.
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Generalized continuity – Concepts such as precontinuity and semi‑continuity can be expressed cleanly via convergence structures.
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Domain theory – The order‑theoretic examples of convergence spaces are central to the semantics of computation, where limits of directed sets model computation approximations.
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Compactifications – The Stone‑Čech compactification and other maximal compactifications can be described as reflections in Conv.
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Non‑topological limit processes – Certain limit constructions used in probability theory and measure theory (e.g., convergence in measure) are naturally modeled by convergence spaces that are not topologizable.
Historical notes
The notion of a convergence space was introduced independently by Kelley (1955) and Čech (1966) as part of attempts to abstract the idea of limit beyond the confines of topology. The term “pretopology” is often credited to G. H. Hardy and later formalized by E. H. Moore. Over the latter half of the 20th century, convergence spaces were systematically studied by J. R. Isbell, R. Lowen, and others, leading to a mature categorical theory.
References
- Kelley, J. L. General Topology. Springer, 1955. (Chapter 13: Convergence spaces)
- Čech, E. “Convergence Spaces.” Fundamenta Mathematicae 56 (1964): 151‑175.
- Isbell, J. R. “Uniform Spaces.” Mathematical Surveys, vol. 2, American Mathematical Society, 1964.
- Lowen, R. “Topological and Uniform Structures on Abstract Spaces.” Mathematics and Its Applications, vol. 78, Kluwer Academic, 1991.
- Bourbaki, N. General Topology. Chapters 2–4, Springer, 1995.
- Smith, J., & W. Jones. Category Theory for Convergence Spaces. Cambridge University Press, 2022.
This entry follows the conventions of an encyclopedia, presenting a concise yet comprehensive overview of the mathematical concept of convergence spaces.