nLab
vanishing at infinity

Contents

Contents

Idea

A map ff between spaces (say, a continuous map between topological spaces) vanishes at infinity if f(x)f(x) gets arbitrarily close to zero as xx gets sufficiently close to infinity.

For a map f:XYf\colon X \to Y, we need a notion of being close to 00 in YY, so take YY to be a pointed space; then getting arbitrarily close to 00 means entering any neighbourhood of the basepoint. We also need a notion of being close to infinity in XX, so take XX to be a locally compact Hausdorff space; then getting sufficiently close to infinity means entering the exterior of some compact subspace. (To interpret ‘getting’, of course, we may use nets.) It is likely, however, that further generalisations are possible.

Definitions

Let XX and YY be topological spaces, and let ff be a continuous map (or potentially any function) from XX to YY. Let YY be pointed, and let XX be locally compact Hausdorff.

Definition

The map f:XYf\colon X \to Y vanishes at infinity if for every neighbourhood NN of the basepoint in YY, there is compact subspace KK of XX such that f(x)f(x) belongs to NN whenever xx lies in the exterior of KK in XX.

In case YY is a pointed metric space (such as a Banach space, with basepoint 00; or in particular the real line, with basepoint 00), then we may equivalently say:

(Here, {\|{-}\|} is the norm in a Banach space, or more generally the distance from the basepoint in any pointed metric space.)

Properties

Relation to compactifications

One way of considering this definition is that one can adjoin to XX a point “at infinity”, denoted \infty, by declaring that the open neighborhoods of \infty are sets of the form {}Ext(K)\{\infty\} \cup Ext(K) for KXK \subset X compact. This is called the one-point compactification, denoted X +X^+. Then a map f:XAf\colon X \to A vanishes at infinity iff there exists an extension of ff to a map X +AX^+ \to A, continuous at \infty (at least) that sends \infty to 00 (thus literally “vanishing at \infty”).

References

Last revised on November 8, 2018 at 03:12:30. See the history of this page for a list of all contributions to it.