For any
ordinal number λ one can consider the spaces of ordinal numbers :[0,\lambda) = \{\alpha \mid \alpha :[0,\lambda] = \{\alpha \mid \alpha \le \lambda\} together with the natural order topology. These spaces are called
ordinal spaces. (Note that in the usual set-theoretic construction of ordinal numbers we have
λ = [0,
λ) and
λ + 1 = [0,
λ]). Obviously, these spaces are mostly of interest when
λ is an infinite ordinal; for finite ordinals, the order topology is simply the
discrete topology. When
λ = ω (the first infinite ordinal), the space [0,ω) is just
N with the usual (still discrete) topology, while [0,ω] is the
one-point compactification of
N. Of particular interest is the case when
λ = ω1, the set of all countable ordinals, and the
first uncountable ordinal. The element ω1 is a
limit point of the subset [0,ω1) even though no
sequence of elements in [0,ω1) has the element ω1 as its limit. In particular, [0,ω1] is not
first-countable. The subspace [0,ω1) is first-countable however, since the only point in [0,ω1] without a countable
local base is ω1. Some further properties include • neither [0,ω1) or [0,ω1] is
separable or
second-countable • [0,ω1] is
compact, while [0,ω1) is
sequentially compact and
countably compact, but not compact or
paracompact == Topology and ordinals ==
Ordinals as topological spaces Any
ordinal number can be viewed as a topological space by endowing it with the order topology (indeed, ordinals are
well-ordered, so in particular
totally ordered). Unless otherwise specified, this is the usual topology given to ordinals. Moreover, if we are willing to accept a
proper class as a topological space, then we may similarly view the class of all ordinals as a topological space with the order topology. The set of
limit points of an ordinal
α is precisely the set of
limit ordinals less than
α.
Successor ordinals (and zero) less than
α are
isolated points in
α. In particular, (meaning the set ) is a
discrete topological space, as is each finite ordinal, but no ordinal greater than is discrete. The ordinal
α is
compact as a topological space if and only if
α is either a successor ordinal or zero. The
closed sets of an ordinal
α are those that contain a limit ordinal less than
α whenever every open interval that contains the limit ordinal intersects the set. Any ordinal is, of course, an open subset of any larger ordinal. We can also define the topology on the ordinals in the following
inductive way: 0 is the empty topological space,
α+1 is obtained by taking the
one-point compactification of
α, and for
δ a limit ordinal,
δ is equipped with the
inductive limit topology. Note that if
α is a successor ordinal, then
α is compact, in which case its one-point compactification
α+1 is the
disjoint union of
α and a point. As topological spaces, all the ordinals are
Hausdorff and even
normal. They are also
totally disconnected (connected components are points),
scattered (every non-empty subspace has an isolated point; in this case, just take the smallest element),
zero-dimensional (the topology has a
clopen basis: here, write an open interval (
β,
γ) as the union of the clopen intervals (
β,
γ'+1) = [
β+1,
γ'] for
γ'1 and its successor ω1+1 are frequently used as textbook examples of uncountable topological spaces. For example, in the topological space ω1+1, the element ω1 is in the closure of the subset ω1 even though no (countably-long) sequence of elements in ω1 has the element ω1 as its limit: an element in ω1 is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one. The space ω1 is
first-countable but not
second-countable, and ω1+1 has neither of these two properties, despite being
compact. It is also worthy of note that any
continuous function from ω1 to
R (the
real line) is eventually constant: so the
Stone–Čech compactification of ω1 is ω1+1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is much
larger than ω).
Ordinal-indexed sequences If
α is a limit ordinal and
X is a set, an
α-indexed sequence of elements of
X merely means a function from
α to
X. This concept, a
transfinite sequence or
ordinal-indexed sequence, is a generalization of the concept of a
sequence. An ordinary sequence corresponds to the case
α = ω. If
X is a topological space, we say that an
α-indexed sequence of elements of
X converges to a limit
x when it converges as a
net, in other words, when given any
neighborhood U of
x there is an ordinal
β ι is in
U for all
ι ≥
β. Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω1 is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable. However, ordinal-indexed sequences are not powerful enough to replace nets (or
filters) in general: for example, on the
Tychonoff plank (the product space (\omega_1+1)\times(\omega+1)), the corner point (\omega_1,\omega) is a limit point (it is in the closure) of the open subset \omega_1\times\omega, but it is not the limit of an ordinal-indexed sequence of ordinals in \omega_1\times\omega. == See also ==