MULTIPLE PUBLIC DEFINITIONS ?

"int a" tells the compiler to reserve space for an integer.

Close, yet not quite completely correct. The statements

int a = 0;
int b = some_value;

do indeed and unquestionably reserve space for an integer (and initialize it), right then and there. But the case of

int a;

without an initializer is trickier.

This is what the standard calls a tentative definition. That's to say it is interpreted as just a declaration, i.e. as if you had written

extern int a;

until the compiler has finished with the source file in question. If, and only if there was no initialized definition of the object found until then, the tentative definition is turned into a real definition, and will actually reserve space. That's how it comes that

int a;
int a;
int a = 5;

is allowed, while

int a = 5;
int a = 5;

isn't.

And to make matters worse, the C language definition leaves it open to the implementor to decide what happens if two separate compilation units (vulgo: source files) both contain only tentative definitions of a same-named object. Most Unix-born C toolchains out there will happily accept a program that has

int a;

in a header #include'd by several source files --- they'll just join all these tentative definitions into a single object, thus extending the tentative status of the definition beyond the source file's compile time, until the time the whole program is linked.

"int a" tells the compiler to reserve space for an integer.

Close, yet not quite completely correct. The statements

int a = 0;
int b = some_value;

do indeed and unquestionably reserve space for an integer (and initialize it), right then and there. But the case of

int a;

without an initializer is trickier.

This is what the standard calls a tentative definition. That's to say it is interpreted as just a declaration, i.e. as if you had written

extern int a;

until the compiler has finished with the source file in question. If, and only if there was no initialized definition of the object found until then, the tentative definition is turned into a real definition, and will actually reserve space. That's how it comes that

int a;
int a;
int a = 5;

is allowed, while

int a = 5;
int a = 5;

isn't.

And to make matters worse, the C language definition leaves it open to the implementor to decide what happens if two separate compilation units (vulgo: source files) both contain only tentative definitions of a same-named object. Most Unix-born C toolchains out there will happily accept a program that has

int a;

in a header #include'd by several source files --- they'll just join all these tentative definitions into a single object, thus extending the tentative status of the definition beyond the source file's compile time, until the time the whole program is linked.