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**Iondor** A property that is definable by using only the predicate "is an open set", the relation "is an element of" and all the set operations arbitrary union, complement and arbitrary intersection, the subset relation etc.

The definition may not refer to any other representation dependent structure besides that.

A homeomorphism preserves these predicates and therefore all properties that are definable with these predicates.

For example the boundary of a set is equivalent to this definition

X is the boundary of S

iff

X is the largest set, such that

for all open U:

$\displaystyle U \cap X \neq \emptyset \rightarrow U\cap S \neq \emptyset \wedge U\cap S^{c}\neq \emptyset$

One could translate this definition into a formula of predicate logic $\displaystyle \phi(X,S)$, which uses only the above mentioned predicates. Homeomorphisms preserve such formulas, that means if f is homeomorphism and $\displaystyle \phi(X,S)$ is true, then also $\displaystyle \phi(f(X),f(S))$ is true.

So if X is the boundary of S, then f(X) is the boundary of f(S).