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Correspondence is a mathematical term used to describe a relationship between two sets. It is represented as a mapping or a function where each element of one set (

• Denoted as the domain, let's say set X, maps to one or more elements of another set
• The range, let's say set Y

The focus is not on the function being deterministic, as in classic functions, but rather on the possible set of outputs for each input.
In our context, correspondences are denoted by functions like \( f: X \rightarrow \mathfrak{B}(Y) \), where \( \mathfrak{B}(Y) \) is the power set of Y, or the set of all possible subsets of Y.
This term is crucial because it helps us understand complex relationships where each input can relate to multiple possible outputs, such as preferences in economic utility functions or choice sets in decision-making rules.

Composite functions occur when two functions are combined to form a new function. If you have two functions, \( f \) and \( g \), their composition \( h \) is defined as \( h(x) = g(f(x)) \). In the realm of correspondences, we need to slightly adjust this definition.
Since each part of the correspondence leads to a set, the composition is the union of all possible output sets. In our example, \( h(x) = \bigcup_{y \in f(x)} g(y) \). This means:

• For an input \( x \), we first apply the correspondence \( f \) to obtain a subset of Y.
• Then, we take each element \( y \) in this subset and apply the function \( g \) to each.
• The result is a union of all the images \( g(y) \).

This results in a total set of possible outcomes for the input \( x \). What's interesting is that when dealing with upper hemicontinuous functions, composing them maintains this property. This is because the union of upper hemicontinuous sets, under specific conditions, will also be upper hemicontinuous.

An open set is a fundamental concept in topology, a branch of mathematics dealing with space, continuity, and dimension. Intuitively, an open set is a set where 'around' every point in the set, there is no boundary, and each point has other points in the set within its neighborhood.

• For instance, in the real numbers on a number line, the interval \( (a, b) \) represents an open set.
• This is because numbers very close to the endpoints \( a \) and \( b \) still lie within the set, yet the endpoints themselves are not included.

These sets play a crucial role when discussing properties like continuity and hemicontinuity. Particularly in upper hemicontinuity,
the definition relies on the ability to talk about neighborhoods in terms of open sets. If for an open set \( U \), the condition \( f(x) \cap U = \emptyset \) implies the same for all points \( x' \) near \( x \), then the correspondence is said to be upper hemicontinuous.
This characteristic of open sets allows mathematicians to express complex ideas about function behavior in more intuitive terms.