In mathematics, there are many senses in which a sequence or a series is said to be convergent.
This article describes various modes (senses or species) of convergence in the settings where they are defined.
Convergence can be defined in terms of sequences in first-countable spaces.
Cauchy nets and filters are generalizations to uniform spaces.
The concept of completeness of metric spaces, and its generalizations is defined in terms of Cauchy sequences.
The most basic type of convergence for a sequence of functions (in particular, it does not assume any topological structure on the domain of the functions) is pointwise convergence.
It is defined as convergence of the sequence of values of the functions at every point.
Pointwise convergence implies pointwise Cauchy convergence, and the converse holds if the space in which the functions take their values is complete.
Roughly speaking, this is because "local" and "compact" connote the same thing.
For functions taking values in a normed linear space, absolute convergence refers to convergence of the series of positive, real-valued functions
For functions defined on a topological space, one can define (as above) local uniform convergence and compact (uniform) convergence in terms of the partial sums of the series.
And if the domain is locally compact (even in the weakest sense), then local normal convergence implies compact normal convergence.
If one considers sequences of measurable functions, then several modes of convergence that depend on measure-theoretic, rather than solely topological properties, arise.