Convergence space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain properties relating elements of X with the <a href="/facts/Family_of_sets/WbYof7lP">family</a> of <a href="/facts/Filter_(set_theory)/7f3zk55u">filters</a> on X. Convergence spaces generalize the notions of <a href="/facts/Filters_in_topology/5vQbHcpE">convergence</a> that are found in <a href="/facts/Point-set_topology/ezC7B0yP">point-set topology</a>, including <a href="/facts/Metric_space/gnBBRXtc">metric convergence</a> and <a href="/facts/Uniform_convergence/ecz9eNWn">uniform convergence</a>. Every <a href="/facts/Topological_space/v2ut7CbK">topological space</a> gives rise to a canonical convergence but there are convergences, known as non-topological convergences, that do not arise from any topological space. An example of convergence that is in general non-topological is <a href="/facts/Almost_everywhere/1quRFtJE">almost everywhere</a> convergence. Many <a href="/facts/Topological_properties/vbJR18Sx">topological properties</a> have generalizations to convergence spaces.
Besides its ability to describe notions of convergence that <a href="/facts/Topology_(structure)/v2ut7CbK">topologies</a> are unable to, the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of convergence spaces has an important categorical property that the <a href="/facts/Category_of_topological_spaces/qH4SaEWa">category of topological spaces</a> lacks. 
The category of topological spaces is not an <a href="/facts/Exponential_object/4MYWRF9h">exponential category</a> (or equivalently, it is not <a href="/facts/Cartesian_closed/mPXAQps2">Cartesian closed</a>) although it is contained in the exponential category of pseudotopological spaces, which is itself a <a href="/facts/Subcategory/k4lEeDDb">subcategory</a> of the (also exponential) category of convergence spaces.

Convergence space open-in-new

Convergence space