Justification for our sequence notation

We are interested in using sequences in a specific way: to identify boundary points. Thus we only care about the limit points of sequences under some envelopment and can restrict our attention to those sequences in _M with no limit points in _M. Taking the set of all such sequences Σ0(M) we then view each sequence, s, under each envelopment φ : M → Mφ ∈ Φ(M) and look at the limit points of φs. We

consider the sequence s to ‘identify’ its limit points under the embedding φ. By construction these limit points must lie on ∂φ_M. As we show below this type of limit point is independent of the ordering of the elements of s. They are in fact accumulation points of the underlying set of the sequenceφsconsidered as a subset of Mφ. Within this section we give the necessary results and definitions to prove

these claims, within the more general setting of a topological space T.

Strictly speaking a sequence is a subset s _{⊂ T} of some topological space _T and a surjection f : N _→ s; we shall denote this by (s, f). As mentioned above, we are only interested in sequences (s, f) that have no limit points inT. As we show below if the set s is countably infinite and has no accumulation points then there exists a bijective function f : N _→ s so that the sequence (s, f) has no limit points in

T. In addition a sequence (s, f) has no limit points, in _T, only if s is countably infinite and has no accumulation points. We go on to show that for any countably infinite subset of T, without accumulation points, and for any continuous function

φ : T → Tφ, where Tφ is first countable, the accumulation points of φ(s) and the

limit points of (φ(s), φ_◦ f), in _Tφ, are the same. Thus sequences (s, f) with no

limit points in _T can be treated as sets. To emphasise this we denote such sets, considered as sequences, using gothic font. That is, we shall write s rather than s. On occasion, when constructing new sequences from old ones, we shall need to refer to specific elements of s. In this case we shall writes={si} for the sequence (s, f)

with f(i) =si.

5.1.1 Definition. Let (s, f) be a sequence in some topological space T. A subse- quence (w, g) of (s, f) is defined as a pair (w, g) so thatw⊂s and for all x, y ∈w,

f−1(x)< f−1(y) impliesg−1(x)< g−1(y). That is the ordering of the elements ofw

5.1 Binary operations on Σ0(M) and their properties 65

5.1.2 Definition. Let (s, f) be a sequence in _T, then x _{∈ T} is a limit point of (s, f) if, for all open U _{∈ N}(x), there exists a countably infinite subset N _⊂ N so that n _∈N implies f(n)∈U.

5.1.3 Definition. A point x_{∈ T} is an accumulation point of a setS _{⊂ T} if, for all open U _{∈ N}(x), (U ₋x)∩S ₆=∅.

5.1.4 Proposition. Let s _{⊂ T} be a countably infinite set with no accumulation points, then there exists a bijection f : N _→ s so that the sequence (s, f) has no limit points.

Proof. Since s is countably infinite we know that there exists a bijective function

f :N _→s. If there existed x a limit point of (s, f), then as s has no accumulation points we can choose an openU _{∈ N}(x) so that (U₋x)_∩s =∅. Asxis a limit point of (s, f), there must exist some countably infinite N _⊂ N so that n _∈ N implies

f(n) _∈ U. We can conclude that n _∈ N implies f(n) = x. Choosing v, u _∈ N we know that f(v) = f(u), but f is bijective and therefore u = v. This provides us with a contradiction as N is countably infinite.

5.1.5 Lemma. Let T be a first-countable topological space. If x _{∈ T} is an accu- mulation point of a countably infinite set s _{⊂ T} then, for all surjective functions

f :N_→s, x is a limit point of (s, f).

Proof. Since_T is first-countable, there exists a countable neighbourhood basis,_{Ui}

at x. We may choose this basis so that if i > j, then Ui ⊂ Uj. For each i, choose

xi ∈ (Ui −x)∩s. Let f : N → s be a surjection, then for each xi, there exists

sj = f(j) so that xi = f(j). Let h : N → N be the function defined by h(i) = j.

That is h is the function that picks out the subset of N which under f can be identified with {xi}. Define a surjection g : N → {xi} by g(i) = f ◦h(i). Thus

we can see that g(i) = f _◦h(i) = f(j) = sj = xi. With this definition, it is clear

that (_{xi}, g) is a subsequence of (s, f). Since, by construction, xis a limit point of

(_{xi}, g) we can conclude that x is a limit point of (s, f) as required.

5.1.6 Proposition. Let (s, f) be a sequence in some topological space _T that has no limit points in _T and let φ :_{T → T}φ be a continuous map of topological spaces,

where Tφ is first-countable. Then the set of limit points of(φ(s), φ◦f)in Tφ is equal

Proof. Let x _{∈ T}φ be a limit point of (φ(s), φ◦f) and suppose that x is not an

accumulation point. Then there exists an open U _{∈ N}Tφ(x) so that (U−x)∩φ(s) =

∅. Asxis a limit point, however, there exists a countably infinite setN _⊂Nso that

n _∈N implies that φ_◦f(n)∈U. Since φ_◦f(n)∈φ(s) and (U ₋x)∩φ(s) =∅ we can conclude that n _∈N implies that φ_◦f(n) = x. As (s, f) has no limit points in

T we know thatx_∈∂φ_T but x=φ_◦f(n)_∈φ_T. This is a contradiction, however, and therefore x must be an accumulation point ofφ(s).

From lemma 5.1.5 we already know that an accumulation point is a limit point.

Of course, all of this requires that our topological spaces _T are first-countable. We note, however, that through the use of filters it may be possible to generalise the results of this section to arbitrary topological spaces and therefore generalise the results that follow (as we only make use of first-countability in this section). For our purposes, we do not need such generality and, within this chapter, we only want to illustrate the generality within which one can define Abstract Boundary-like sets; thus we shall not explore the details of filters here.