Support (mathematics)

The support of a function is the closure of the set where it is nonzero. For a distribution, it is the smallest closed set outside which the distribution assigns probability zero.

Closure (topology): The smallest closed set containing a given subset of a topological space. Taking the closure makes function support closed even when nonzero points omit boundary points.
Zero set: The set of inputs at which a function takes the value zero. Function support is the complement of the interior of its zero set.
Partition of unity: A collection of smooth functions whose sum is one and whose supports are locally finite. Controlled supports let local constructions combine into global ones.
Pointwise convergence: A sequence of functions converges pointwise when it converges at each input separately. Pointwise limits can have supports unlike those suggested by supports of the approximating functions.
Open set: A subset of a topological space whose points each have a neighborhood contained in the subset. The complement of support is the largest open region where a function vanishes.
Essential support: The smallest closed set outside which a measurable function is zero almost everywhere. It ignores changes on measure-zero sets that ordinary pointwise support detects.
Convolution: An operation combining two functions or measures by integrating products over translated inputs. The support of a convolution is constrained by sums of the input supports.
Almost everywhere: A property holds almost everywhere when its exceptions form a set of measure zero. Essential support treats functions equal almost everywhere as having the same support.
Continuous function: A function whose nearby inputs always have nearby outputs, expressed through the topology of its domain and codomain. Continuity makes the nonzero set open, clarifying how its closure forms support.
Support of a distribution: The complement of the largest open set on which a distribution vanishes. This definition extends support beyond ordinary functions to generalized functions.

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