In some books the word “if” is used in definitions and it is not clear if they actually mean “iff” (i.e “if and only if”).
I’d like to know if in mathematical literature in general “if” in definitions means “iff”.
For example I am reading “Essential topology” and the following definition is written:
In a topological space T, a collection B of open subsets of T is said
to form a basis for the topology on T if every open subset of T can be written as a union of sets in B.
Should I assume the converse in such a case?
Should I assume that given a basis B for a topological space, every open set can be written as a union of sets in B?
This is just an example, I am not asking specifically about this sentence.
As it is a definition, the validity of the property (here “being a basis for the topology”) must be defined by it in all cases. So implicitly all cases not mentioned do not have the property.
This convention is even stronger than the “if” meaning “iff” in definitions. Take as example the definition “an integer p>1 is called a prime number if it cannot be written as a product p=ab of integers a,b>1“. This says that 6 is not a prime number, since 6=2×3; this is an instance of the “iff” meaning in definitions. But it also says implicitly that 1 is not a prime number, nor −5 nor π nor exp(πi/3) nor GL(3,R), as none of these can be described as integers p>1; even a statement with “iff” would in itself not seem to state non-primality of those objects. But since it is a definition, anything that does not match its description is implicitly excluded from the property.
Without this implicit exclusion of cases not mentioned, it would be very hard indeed to give a complete definition of any property. Imagine (assuming the “everything is a set” philosophy) the ugliness of “a set x is called a prime number if x∈Z and x>1 and …”.