In mathematics, an axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist.
σ-compact spaces: there exists a countable cover by compact spaces,
These axioms are not all unrelated. In particular, every second-countable space is first-countable, separable, and Lindelöf. Also, every σ-compact space is Lindelöf. For metric spaces, first-countability is automatic, and second-countability, separability, and the Lindelöf property are all equivalent.
In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results.
Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups).
The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four.
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