It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
Provided X has at least two elements, this is equivalent to saying that the open intervals together with the above rays form a base for the order topology.
The order topology makes X into a completely normal Hausdorff space.
Under the subspace topology, the singleton set {−1} is open in Y, but under the induced order topology, any open set containing −1 must contain all but finitely many members of the space.
Assume without loss of generality that −1 is the smallest element of Z.
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 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 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).
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.
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'<γ).
The space ω1 is first-countable but not second-countable, and ω1+1 has neither of these two properties, despite being compact.
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 < α such that xι 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
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