Vanish at infinity

In mathematics, a function on a normed vector space is said to vanish at infinity if

For example, the function

defined on the real line vanishes at infinity.

More generally, a function ${\displaystyle f}$ on a locally compact space (which may not have a norm) vanishes at infinity if, given any positive number ${\displaystyle \epsilon }$, there is a compact subset ${\displaystyle K}$ such that

whenever the point ${\displaystyle x}$ lies outside of ${\displaystyle K}$.
In other words, for each positive number ${\displaystyle \epsilon }$ the set ${\displaystyle \left\{x\in X:\|f(x)\|\geq \epsilon \right\}}$ is compact.
For a given locally compact space ${\displaystyle \Omega }$, the set of such functions

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