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
A continuous map between topological spaces vanishes at infinity if gets arbitrarily close to zero as gets sufficiently close to infinity.
In order to make sense of this for a map , one:
assumes that is pointed by an element ,
then getting arbitrarily close to means entering any neighbourhood of this basepoint.
assumes that is locally compact Hausdorff,
then getting sufficiently close to infinity means entering the exterior of some compact subspace .
Finally “getting close” is understood in terms of nets.
Consider:
a point topological spaces, with basepoint to be denoted ,
The map vanishes at infinity if for every neighbourhood of the basepoint , there is a compact subspace of such that for all in the exterior of , we have .
If is a pointed metric space (such as a Banach space, in particular the real line), then Def. is equivalent to:
Here, denotes the distance from the basepoint in the metric space, such as the norm in the case of Banach spaces.
One may neatly understand Def. as saying that a function “vanishes at infinity” if it literally sends “”, where “” denotes the point adjoined to when passing to its “one-point compactification” (and demanding that extends to a continuous function on ).
This is achieved by the fact that in — whose underlying set is — the open neighborhoods of are subsets of the form for compact .
Precisely:
A continuous function vanishes at infinity according to Def. iff it extends along to a continuous map such that .
For applications of this equivalence to physics see:
In view of the equivalence of Prop. , we may say that for locally compact , the vector space of continuous maps that vanish at , denoted , is isomorphic to the kernel of the evaluation map
from the function space on , with pointwise defined -algebra structure and the sup-norm topology.
Since the function space is a -algebra and is a closed subspace, and thus a Banach space, and since the -algebra structure on clearly restricts to a -algebra structure on the kernel, the result is clear.
is nonunital unless is already compact. Its unitalization is .
The algebra is discussed in most monographs on -algebras, for instance:
Garth Warner Ex. 1.15 in: -Algebras, EPrint Collection, University of Washington (2010) [hdl:1773/16302, pdf, pdf]
Bruce Blackadar Ex. II.1.1.3(ii) in: Operator Algebras – Theory of -Algebras and von Neumann Algebras, Encyclopaedia of Mathematical Sciences 122, Springer (2006) [doi:10.1007/3-540-28517-2]
(which however does not dwell on the definition of “vanishing at infinity”)
Ian Putnam, Ex. 1.2.4 in: Lecture notes on -algebras (2019) [pdf, pdf]
See also:
Wikipedia: Vanish at infinity
Wikipedia: Locally compact space
Last revised on June 27, 2025 at 08:26:43. See the history of this page for a list of all contributions to it.