coercive function

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coercive function

Definition 1 (coercive function).

Let $X$ and $Y$ be topological spaces. A function $f\colon X\to Y$ is said to be coercive if for every compact set $J\subset Y$ there exists a compact set $K\subset X$ such that

F(X\setminus K)\subset Y\setminus J.

The general definition given above has a clear sense when specialized to the Euclidean spaces, as shown in the following result.

Proposition 1 (coercive functions on $\mathbb{R}^{n}$ ).

A function $f\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ is coercive if and only if

\lim_{{|x|\to+\infty}}|f(x)|=+\infty.

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Mathematics Subject Classification

54A05 Topological spaces and generalizations (closure spaces, etc.)

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coercive function

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coercive function

Definition 1 (coercive function).

Proposition 1 (coercive functions on $\mathbb{R}^{n}$ ).

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coercive function

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Definition 1 (coercive function).

Proposition 1 (coercive functions on Rnsuperscriptnormal-Rn\mathbb{R}^{n}).

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Proposition 1 (coercive functions on $\mathbb{R}^{n}$ ).