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Injective and functional are completely standard in this case. This is what you should use. The term "functional" is not overloaded, when you are using it to say that something is a function. Being fu …

In spaces where singleton points are closed, your property is equivalent to saying that the space has no isolated points. Or in other words, that it is perfect. Clearly, no space with an isolated po …

Having needed this concept in a recent article, I used the term reduct substructure in exactly this situation, but I haven't seen this terminology elsewhere and I don't think there is an established terminology …

Let me mention as a counterpoint that there is less need for new terminology than one might expect. … Mathematical exposition is often more successful and clearer without new terminology, and one should consider whether one needs any new terminology at all. …

A partial order with no infinite descending chains is said to be well-founded. Every well-founded partial order admits an ordinal ranking function, an assignment of nodes in the order to ordinals, suc …

Define an order on pairs of ordinals $(\alpha,\beta)$ by ordering first by maximum, then by first coordinate, then by second coordinate. That is, one pair preceeds another if the maximum is smaller, o …

This is called an $L$-valued relation, when $L$ is the target of the function, which can be viewed as the collection of possible truth values. Thus, a $2$-valued relation is just an ordinary relatio …

The subscript $0$ is meant to indicate the amount of induction that the theory has. The wikipedia entry on Reverse mathematics says of the big five theories of reverse mathematics that The …

The elements of this partition are precisely the atoms of the complete Boolean algebra generated by the family.

I like your concept a lot, and have been able to find a characterization. Suppose that $f:N\to N$ is effectively closed in your sense. First, as you mentioned, it is easy to see that $\text{ran}(f)$ …

The term "class" is not a technical term with a universally definite meaning, but there are various established meanings in various contexts. In ZFC the established usage as Wojowu mentions in the com …

In weak set theories, using classical logic and interpreting "small subclass" as "set", this principle amounts to an alternative formulation of the collection axiom. For example, in Zermelo set theory …

The answer is no for uncountable cardinals $\kappa$. Let $\mathfrak{A}=\langle A,U\rangle$ be a set $A$ of size $\kappa$ with a unary predicate $U\subset A$, where $U$ and $A-U$ both have size $\kappa …

Since you are taking the complement of a substring, and it appears that there may be no firmly established terminology, I propose: a substring complement is what remains after deleting a substring, … I would prefer this natural language terminology over the alternative co-substring and co-subsequence, which sound unnecessarily technical to my ear, but this difference may be slight. …

In that terminology, trees of your first kind are known as the well-founded trees, since they are trees where the tree order is well-founded (and well-founded linear orders are the same as well-orders) … I think that the situation is that because set theorists are mainly interested in the well-founded case, the terminology evolved to drop the adjective from well-founded trees. …