is
ultraconnected, since the closures of the singletons and contain the product as a common element.
Hence is a
normal space. But is not
completely normal. For example, the singletons and are
separated sets (6 is not a multiple of 4 and 4 is not a multiple of 6), but have no disjoint open neighborhoods, as their smallest respective open neighborhoods meet non-trivially in .
is not a
regular space, as a basic neighborhood is finite, but the closure of a point is infinite.
is a
scattered space, as each nonempty subset has a first element, which is an isolated element of the set.
The
compact subsets of are the finite subsets, since any set is covered by the collection of all basic open sets , which are each finite, and if is covered by only finitely many of them, it must itself be finite. In particular, is not
compact.
is
locally compact in the sense that each point has a compact neighborhood ( is finite). But points don't have closed compact neighborhoods ( is not
locally relatively compact.)
is
ultraconnected, since the closures of the singletons and contain the product as a common element.
Hence is a
normal space. But is not
completely normal. For example, the singletons and are
separated sets (6 is not a multiple of 4 and 4 is not a multiple of 6), but have no disjoint open neighborhoods, as their smallest respective open neighborhoods meet non-trivially in .
is not a
regular space, as a basic neighborhood is finite, but the closure of a point is infinite.
is a
scattered space, as each nonempty subset has a first element, which is an isolated element of the set.
The
compact subsets of are the finite subsets, since any set is covered by the collection of all basic open sets , which are each finite, and if is covered by only finitely many of them, it must itself be finite. In particular, is not
compact.
is
locally compact in the sense that each point has a compact neighborhood ( is finite). But points don't have closed compact neighborhoods ( is not
locally relatively compact.)