This article is about support of a function. For other notions of support, see support.
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Given a pointed set with specified element , a set , and a function , the support of is the subset of on which is not equal to .
In constructive mathematics, there are multiple notions of inequality, due to the failure of the double negation law. As a result, there are multiple notion of support of a function. Thus, we define the following:
Given a pointed set with specified element , a set , and a function , the support of is the subset of on which is not equal to .
Given a pointed set with an tight apartness relation and specified element , a set , and a function , the strong support of is the subset of on which is apart from .
In topology the support of a continuous function as above is the topological closure of the set of points on which does not vanish:
If is a compact subspace, then one says that has compact support.
Last revised on October 17, 2022 at 16:52:47. See the history of this page for a list of all contributions to it.