User:InfCompact/Semi-compact function

In mathematical analysis and optimization, a semi-compact function generalizes the conditions needed in the Extreme value theorem for the function to attain its optimal values.

Formal definition
Let $$X$$ be a topological space.

A function $$f : X \to \mathbb{R}$$ is upper semi-compact (or sup-compact) if for all $$\lambda \in \mathbb{R}$$ the superlevel set $$\{x \in X : f(x) \geq \lambda\}$$ is compact.

A function $$f$$ is lower semi-compact (or inf-compact) if for all $$\lambda \in \mathbb{R}$$ the sublevel set $$\{x \in X : f(x) \leq \lambda\}$$ is compact.

Properties
The main implication of semi-compactness is the attainment of optimal values. This generalizes the Extreme value theorem.

Theorem. ''Let $$X$$ be a Hausdorff topological space, and let $$f : X \to \mathbb{R}$$ be upper semi-compact. Then there exists $$x^\star \in X$$ such that $$f(x^\star) = \sup_{x \in X} f(x)$$.''

Proof. Let $$s = \sup_{x \in X} f(x)$$. We observe that the superlevel sets $$L_n = \{x \in X : f(x) \leq s - 2^{-n}\}$$ are all compact by definition of upper semi-compactness. Since $$X$$ is Hausdorff, these sets are also closed, and they satisfy the finite intersection property. Therefore, $$L = \bigcap_{n=1}^{\infty} L_n = \{x \in X : f(x) = s\}$$ is a nonempty set, and so there exists $$x^\star \in L$$ such that $$f(x^\star) = \sup_{x \in X} f(x)$$. $$\square$$

Similarly, lower semi-compact functions attain their minima.

Relation to semi-continuity
In general topological spaces, semi-continuity and semi-compactness are independent concepts. In a space where compact sets are closed, such as Hausdorff spaces, every upper semi-compact function is upper semi-continuous. Conversely, if $$f : X \to \mathbb{R}$$ is upper semi-continuous and $$X$$ is compact, then $$f$$ is upper semi-compact.