Abstract
We generalise some results of R. E. Stong concerning finite spaces to wider subclasses of Alexandroff spaces. These include theorems on function spaces, cores and homotopy type. In particular, we characterize pairs of spaces such that the compactopen topology on is Alexandroff, introduce the classes of finitepaths and boundedpaths spaces and show that every boundedpaths space and every countable finitepaths space has a core as its strong deformation retract. Moreover, two boundedpaths or countable finitepaths spaces are homotopy equivalent if and only if their cores are homeomorphic. Some results are proved concerning cores and homotopy type of locally finite spaces and spaces of height 1. We also discuss a mistake found in an article of F.G. Arenas on Alexandroff spaces.
It is noted that some theorems of G. Minian and J. Barmak concerning the weak homotopy type of finite spaces and the results of R. E. Stong on finite Hspaces and maps from compact polyhedrons to finite spaces do hold for wider classes of Alexandroff spaces.
Since the category of Alexandroff spaces is equivalent to the category of posets, our results may lead to a deeper understanding of the notion of a core of an infinite poset.
On homotopy types of Alexandroff spaces
Michał Kukieła
Faculty of Mathematics and Computer Science,
Nicolaus Copernicus University,
ul. Chopina 12/18,
87100 Toruń, Poland
Email address:
2000 Mathematics Subject Classification: 06A06, 55P15
Keywords: Alexandroff space, poset, core, compactopen topology, homotopy type, locally finite space
1 Introduction
Alexandroff spaces, first introduced by P. Alexandroff [1], are topological spaces which have the property that intersection of any family of their open subsets is open. Finitie spaces are an important special case. A basic property of the category of Alexandroff spaces (and continuous maps) is that it is isomorphic to the category of preorders (and orderpreserving maps), while the category of Alexandroff spaces is isomorphic to the category of posets.
From the point of view of general topology, Alexandroff spaces seem not to have drawn much interest since Alexandroff’s work (there are, however, some articles on this topic, for example [2] or [13]). On the other hand, they have gained much attention because of their use in digital topology, cf. [11], [16].
The situation is somewhat different if we turn to algebraic topology. Homotopy type and weak homotopy type of Alexandroff spaces were studied by several authors. For finite spaces, these were investigated for example by Stong in [19], Minian and Barmak in [3], [4]. The latter two authors also developed the basics of simple homotopy theory for finite spaces in [5]. In [15] McCord shows quite a surprising result – for every polyhedron exists an Alexandroff space that is weak homotopy equivalent to it (and vice versa). Also [10] contains some results on this theme.
In this article we concentrate mostly on generalising the results of Stong [19], which we shall now shortly describe. Stong’s article contains results concerning homotopies of maps between finite spaces and homotopy type classification of finite spaces via their cores. Cores are also known in order theory (under the same name) and were rediscovered for this discipline by Duffus and Rival [6]. From our point of view, cores are a special kind of deformation retracts. For a finite space, its core may be achieved by removing from the space, one by one, points known as beat points. (Our terminology comes from May’s notes [14], it was also used by Minian and Barmak in [4]. Stong called the points linear and colinear, while in order theory they are called irreducible points. We decided to use the name “beat points” mainly because it is the shortest one.) It turns out that two finite spaces are homotopy equivalent if and only if their cores are homeomorphic. An attempt to generalise these results to locally finite spaces was made by Arenas in [2]. Unfortunately, his paper contains a mistake that makes the theory look a lot simpler than it really is. (Arenas states that with the compactopen topology is Alexandroff for Alexandroff. We show that this is never true if is infinite.) In this article we take roughly the same route, but avoid falling into the same trap.
In Sections 2 and 3 we prepare the ground for further results. Basic facts concerning Alexandroff spaces are recalled and the classes of locally finite spaces, fpspaces and bpspaces are introduced. Section 4 is a study of the compactopen topology on the space of continuous functions between two Alexandroff spaces. We estabilish a bijection between paths in and homotopies. Then we describe how to construct some of these paths and apply the results to show contractibility of the Khalimsky line (also known under the name “infinite fence”). It is also here that the mistake in [2] is discussed. In Section 5 we investigate cores of spaces belonging to the classes introduced ind Section 3. In particular, we show that every fpspace has a core and that two bpspaces or countable fpspaces are homotopy equivalent if and only if their cores are isomorphic. This may be seen as an another approach to the results of Section 3 of Farley’s article [8]. The last section contains some theorems on points (a concept introduced by Barmak and Minian in [4] and related to the weak homotopy type of Alexandroff spaces), Alexandroff Hspaces and maps from compact polyhedrons into Alexandroff spaces. Some of the results presented in this article, especially in Section 5, may be known to ordertheoretists, but they seem not to have been applied to studying homotopy types of infinite Alexandroff spaces.
The author believes the topological approach to cores of infinite posets may yield new insights into their, still not well understood, nature. On the other hand, expressing topological notions, such as homotopy, by the means of the language of order theory greatly simplifies their investigation.
2 Preliminaries
We will now set up some terminology and recall a few basic properties of Alexandroff spaces. Some references for this section are [13], [14], [19], though the latter two articles are concerned with finite spaces only. Nevertheless, the following results may be proved for all Alexandroff spaces the same way it is done in the finite case. The reader is assumed to have some background in order theory, general and algebraic topology.
A topological space is called an Alexandroff space iff arbitrary intersections of sets open in are open. For every the intersection of all open neighbourhoods of is then the minimal open neighbourhood of . We shall denote it by . The family is a basis for the topology on . (Existence of minimal open neighbourhoods of every point in a space is in fact equivalent to that space being Alexandroff.) By we denote the category of Alexandroff spaces and continuous maps and by its full subcategory of Alexandroff spaces.
A preorder is a set equipped with a transitive, reflexive, binary relation. A partially ordered set (a poset) is a preorder such that the binary relation is also antisymmetric. Given two elements of a preorder by we will denote the fact that they are comparable, that is or . A poset is called chaincomplete, if every chain in has both a supremum and an infimum in . For a poset by we denote the set of elements maximal in and by the set of elements minimal in . For we define . Subset of a poset is called a downset if for every . By we denote the category of preorders and orderpreserving maps and by its full subcategory of partially ordered sers.
With every topological space we associate its specialization preorder, that is a preorder such that iff . This association is functorial. Continuous functions are mapped to increasing functions (with the same graph). If is , then its specialization preorder is a partial order. If is , then its specialization preorder is an antichain.
With every preorder we associate an Alexandroff space , with topology generated by the open basis . Therefore, a set is open in iff it is a downset in . is also a functor, increasing functions between preorders are continuous functions between the associated spaces. If is a partial order, then is .
If we restrict to the category , then the functors and are mutually inverse. The same property holds if we exchange with and with . Therefore, in this article we will not make a difference between an Alexandroff space and its associated preorder. The language used will come both from topology and ordertheory.
Starting from this point, all spaces are assumed to be , unless stated otherwise. Since it may be shown the Kolmogorov quotient of an Alexandroff space is homotopy equivalent to that space (this is a consequence of the fact that Lemma 4.13 works also for non spaces), from our point of view little generality is lost, while the exposition gets clearer and shorter.
By we denote the unit interval with the Euclidean topology. The following lemma is a well known result.
Lemma 2.1.
Let be an Alexandroff space and be such that . Then there exists a path with and .
Proof.
Let
If are such that for every , then from the lemma it follows that there exists a path in from to . In particular is path connected and thus Alexandroff spaces are locally path connected. It is straightforward to check that for any the set of such that exists a finite sequence is clopen. Therefore, it is the connected component (and the path component) of .
A Khalimsky line (or a twoway infinite fence) is a space homeomorphic to , where iff either or and is even. A Khalimsky halfline (or a oneway infinite fence) is a space homeomorphic to one of the following two subpaces of : , , where refer to the standard order on . Connected, finite subsets of the Khalimsky line are called finite fences. Such spaces are important both in order theory and in digital topology (where they serve as an analogue of intervals of the real line, cf. [11]).
3 Some classes of Alexandroff spaces
In this section we introduce a few subclasses of Alexandroff spaces and prove some of their properties. Each of these classes is, in a sense, close to the class of finite spaces. Keep in mind that all spaces are assumed .
Definition 3.1.
We call a sequence (finite or infinite) of elements of an Alexandroff space an spath if for and for all . Given a finite spath we say is the length of and call an spath from to .
Definition 3.2.
We say an Alexandroff space is:

a finitechains space, if every chain in is finite,

a locally finite space, if for every the set is finite,

a finitepaths space (fpspace), if every spath of elements of is finite,

a boundedpaths space (bpspace), if exists such that every spath of elements of has less than elements.
Note that different notions of local finiteness are in use, which do not coincide and should not be confused with our definition. For example, sometimes it is only required that is finite for every . Also, the name is sometimes used for posets with the property that all intervals are finite.
Obviously, bpspaces form a (strict) subclass of fpspaces and both fpspaces and locally finite spaces are (strict) subclasses of finitechains spaces. Moreover, from Proposition 3.4 below it follows that connected spaces which are both locally finite and fpspaces must be finite.
Definition 3.3.
Recall from [7] we call a subset of a poset spanning if . By we mean the minimal length of an spath in from to (or if such an spath does not exist). is called isometric if for all . By we denote the set .
Proposition 3.4.
A connected, locally finite space is infinite if and only if it contains an isometric, spanning subset isomorphic to the Khalimsky halfline.
Proof.
The ”if” part is trivial.
For the ”only if” part, suppose is a connected, locally finite space and fix a point . Since is connected, for every exists an isometric fence with . Without lack of generality we may assume for all . Let be given by . For every the set is finite, as a subset of the finite set . Moreover, it is easy to see that the fence
is isometric. Since is locally finite, is finite for every . Thus for any . Therefore, for every exists an with . By the König’s lemma (cf. Chapter III, §5 of [12]) there exists an infinite sequence with for all . It is isometric, which follows from the fact that all fences are isometric. Moreover, we have assumed for all , thus the oneway infinite fence is spanning. ∎
We get a simple corollary.
Corollary 3.5.
An infinite, locally finite space does not have the fixed point property.
Proof.
Definition 3.6.
We say an Alexandroff space satisfies:

the descending chain condition (DCC), if it contains no strictly decreasing infinite sequence,

the ascending chain condition (ACC), if it contains no strictly increasing infinite sequence.
Proofs of the following two lemmas are standard.
Lemma 3.7.
An Alexandroff space is a finitechains space iff it satisfies both ACC and DCC.
Lemma 3.8.
An Alexandroff space satisfies ACC iff for every subset of the set is not empty. (By duality, the same result holds for spaces satisfying DCC and sets .)
4 Function spaces
If are topological spaces, by we denote the set of continuous functions from to equipped with the compactopen topology.
Lemma 4.1 (cf. [9]).
If are topological spaces which satisfy the first countability axiom and is an arbitrary topological space, then continuity of is equivalent to continuity of , where .
The above lemma holds for being an Alexandroff space (each point has a neighbourhood basis consisting of one element – the minimal neighbourhood, so the space is firstcountable) and , the unit interval. Therefore, we have the following corollary.
Corollary 4.2.
Let be an Alexandroff space, an arbitrary topological space. Maps are homotopic if and only if they belong to the same path component of .
On the space , where is Alexandroff and an arbitrary topological space, we define an order iff for all . The following proposition may be proved as Proposition 9 is in [19].
Proposition 4.3.
Let be a topological space and an Alexandroff space. The intersection of all open sets in containing the map is equal to .
Therefore, implies , so the order on defined above is the specialization order of . In Theorem 3.1 of [2] it is stated that is an Alexandroff space, which in general is not true (intersection of all open sets containing a map doesn’t have to be open). We will show this in Theorem 4.10. As a consequence of the mistake, several other untrue statements are made in [2]; we will point them out further in the text.
By , where , , we will denote the set . For brevity, by we denote .
Proposition 4.4.
If is an Alexandroff space, then is compact if and only if is finite and for every exists an such that .
Proof.
For the ”if” part, let be an open cover of . For every choose an open set such that . If for every exists an such that , then (since open sets are downsets) is a subcover of . If is finite, then the subcover is finite.
To prove the reverse implication, let . First suppose that is infinite. Then does not have a finite subcover, since the only set in containing an is , so every subcover of contains an infinite subset . Now suppose there exists an such that for every exists a . If is a finite subcover of , then we can choose an such that , and it is maximal among elements having this property. But then exists a . By choice of , . This is a contradiction with being a cover. ∎
Now, let’s use our characterization of compact sets to examine the compactopen topology on for Alexandroff.
Lemma 4.5.
Let be Alexandroff spaces, be compact, be open. Then .
Proof.
, so . For the reverse inclusion, notice that iff for all . But for every exists an with . Because is order preserving, , and since is a down set, if . Therefore, . ∎
Corollary 4.6.
Let be Alexandroff spaces. The family is a subbasis for the compactopen topology on . Therefore, the compactopen topology on coincides with the topology induced from the product space .
Proof.
From Lemma 4.5 we know that sets of the form , where and is open, form a subbasis for . Because , sets of the form also constitute a subbasis. Now, for a basic set we have:
so is a subbasis for the compactopen topology on .
The second statement of the corollary follows from the above result and the definition of the product topology. ∎
Corollary 4.7.
Let be a finite space, an Alexandroff space. Then the compactopen topology on is Alexandroff.
Proof.
This follows from Corollary 4.6 and the (easilychecked) fact that finite products and subsets of Alexandroff spaces are Alexandroff. ∎
Corollary 4.8.
Let be Alexandroff spaces. Then the compactopen topology on is weaker than the Alexandroff topology induced by order on .
Proof.
Since are downsets and both unions and intersections of downsets are again downsets, open sets in are all downsets and therefore are open in the Alexandroff topology. (This corollary may also be deduced from the general fact that Alexandroff topology is the strongest topology inducing given preorder.) ∎
Lemma 4.9.
An Alexandroff space is hereditarily compact if and only if it contains no infinite antichains and satisfies the ACC.
Proof.
If is hereditarily compact, then it contains no infinite antichains and no infinite, strictly increasing sequences, since these sets are not compact.
To prove the reverse, notice that if contains no infinite antichains and satisfies ACC, then every subset of also has this property. Therefore, it suffices to show that a space satisfying ACC and containing no infinite antichains is compact. By Lemma 3.8, for every . But , so every element of is under some maximal element. If the space also contains no infinite antichains, then the set of its maximal points is finite (because it is an antichain) and the space is compact. ∎
Theorem 4.10.
Let be an infinite Alexandroff space, be an arbitrary Alexandroff space with . Then the following statements are true.

If is discrete, then is Alexandroff if and only if has finitely many connected components.

If is not discrete, then is Alexandroff if and only if is hereditarily compact and for every the image is finite.
Proof.
Since we know that the compactopen topology on is always weaker than the Alexandroff topology, we will only investigate for what spaces it is also stronger, and that is the case if and only if is open for every .
For the proof of 1, suppose is discrete and . Since is continuous, it maps every connected component of to a point. If has finitely many connected components, say , for every component we may choose an . Then , which is open in . Now suppose has infinitely many connected components. We will show that any basic set contains a map that is not in . For every set of this form a connected component of exists such that . Since , there is an such that is not mapped to the point . Let
To prove 2, first assume is not hereditarily compact. Notice that if is not discrete, then it contains a copy of the Sierpiński space (that is the space with topology ) as its subspace. Now, since is not hereditarily compact, a subspace exists that is not compact. Let . Define by and . Then , which means the preimage of the only nontrivial open set in is a downset in . Therefore is continuous. For a basic neighbourhood of , for all . Since is not compact, an exists such that for all . Now define by
Suppose now that is hereditarily compact. If is finite for an , then is an open set. Therefore, if is finite for every , we are finished. Suppose a exists such that is infinite. is hereditarily compact as a continuous image of a hereditarily compact space. Therefore satisfies ACC and has no infinite antichains. It is a wellknown fact that every infinite poset contains either an infinite chain or an infinite antichain (see Corollary 2.5.10 of [18]). is infinite, so it cannot satisfy DCC (otherwise, it would be a finitechains space without infinite antichains, thus finite), which means contains a descending chain . For every choose an . Then is an infinite, hereditarily compact subset of and, by the same reasoning as for , it must contain an infinite descending chain . We will now define a map . For let , for , where , let , and for let . From the construction it is easy to see that is orderpreserving. Now take a basic neighbourhood of . If there exists an , then let be the greatest number such that there exists an . Otherwise, let . Define by
Then is orderpreserving and . ∎
Corollary 4.11.
For an infinite Alexandroff space the space is not Alexandroff.
Proof.
Suppose is an infinite Alexandroff space and is Alexandroff. If was discrete, then by part 1 of Theorem 4.10, would have only finitely many components and therefore it would be finite. Thus is not discrete. But by part 2 of the theorem, it means that every map in has a finite image. In particular is finite. A contradiction. ∎
Corollary 4.12.
If is a hereditarily compact Alexandroff space and is an Alexandroff space satisfying the DCC, then is Alexandroff.
Proof.
For every the image is a hereditarily compact DCC space, thus it contains no infinite chains and no infinite antichains, and, by Corollary 2.5.10 of [18], is finite. ∎
In Corollary 3.3 of [2] it is stated that homotopy classes of maps from to ( being Alexandroff) coincide with connected components of . That would follow if was locally path connected, in particular if was Alexandroff. But, for example, if is an infinite discrete space, then is not locally path connected, so we cannot prove the statement this way. On the other hand, in the given counterexample path components and connected components of are trivially the same. Therefore, we pose the following question.
Question 1.
For Alexandroff spaces , do the path components of coincide with the connected components of ? If it is not so in the general, what conditions on do guarantee such a coincidence?
Using some of the above results on we shall now describe a class of homotopies between functions in (or rather a class of paths in , which is equivalent, since Corollary 4.2 holds). We begin with the following lemma.
Lemma 4.13.
Let be Alexandroff spaces. If are such that for all , then is homotopic to by a homotopy that is constant on the set .
Proof.
If for all , then , where the map is orderpreserving. Therefore, it is sufficient to prove the lemma for .
If , then, by Lemma 2.1, there is a path given by
If we can find such that for , then . If were Alexandroff, all homotopies could be expressed in such a way. In general, however, this is not the case (see Example 4.17).
Theorem 4.14.
Let be Alexandroff spaces. If there exists an infinite sequence of functions such that for all , and a function such that for every exists with the property that for all , then is homotopic to .
Proof.
We may assume . Otherwise, the sequence may be modified like in the proof of Lemma 4.13 to fulfil this condition.
For let denote the path from to such as constructed in Lemma 4.13. Explicitly, is given by
if and
if .
We shall define a path from to . For , let and let . Obviously, is welldefined. To complete the proof, it remains to show that is continuous. It’s easy to see that is continuous at every as a “composition” of paths. Therefore, we only have to show continuity at 1. Let be any subbasic neighbourhood of . That means . We know an exists such that for all , and thus for and , which completes the proof. ∎
As a simple application of Theorem 4.14 we will now characterize a class of contractible spaces (including the Khalimsky line and halfline).
Theorem 4.15.
Let be a finitechains space and a point such that for every exists one and only one spath
with the property that one of the elements is a cover of the other for all . Then is a strong deformation retract of .
Proof.
Let for be given by
Obviously, for all .
We need to show the maps are order preserving. Suppose . There is a finite chain with and . If , then it is easy to see for all . Suppose . One of the spaths , forms the first elements of the other, and the further elements of the longer path come from . Otherwise, we could define another spaths that would have the above property. Suppose . (In the other case we proceed analogously.) If , then . If , then and , thus .
Application of Theorem 4.14 gives a homotopy from to . ∎
Corollary 4.16.
Let be an Alexandroff space of height 1. Then the following statements are equivalent.

has a point as its strong deformation retract.

is path connected and the fundamental group of is trivial.

is path connected and the first homology group of is trivial.

is connected and it contains no crowns.
Proof.
Example 4.17.
Let be the Khalimsky halfline, as described in Section 2. By Corollary 4.16, is contractible. (The same applies to the Khalimsky line.)
On the other hand, it is easy to see that the only continuous function with is given by and if . Now, the only function other than identity is given by and if . Continuing the reasoning we get an infinite sequence of comparable functions, none of which is constant. Therefore, there is no finite sequence with a constant map.
We shall prove a generalisation of Theorem 4.14, which will be useful in the next section.
Theorem 4.18.
Let be Alexandroff spaces and , where is a countable ordinal, be a family of continuous maps such that:

if , then ;

if is a limit ordinal, then for every exists such that for all .
Then is homotopic to .
Proof.
It is a wellknown fact that any countable total order is isomorphic to a suborder of the rational numbers (cf. Chapter VI, §3, Theorem 3 of [12]), and thus of the real unit interval. Therefore, we will identify with a subset of . Moreover, we may assume that and (where the first in this equality indicates the ordinal number and the second is the real number).
We now define . For every ordinal let the restriction be constructed from the path from to such as defined in Lemma 4.13, in the same way it is done in the proof of Theorem 4.14. Let .
It now suffices to show that is continuous. This may be done by a similar argument as in the proof of Theorem 4.14, but applied to every limit ordinal . ∎
5 Cores
We shall now discuss cores of Alexandroff spaces, which from our viewpoint are special kinds of deformation retracts.
Definition 5.1.
Let (or: ) be an Alexandroff space (with the distinguished point ). A point () is called an upbeat point if the set has a smallest element . Dually, is a downbeat point if has a largest element . A point that is either an upbeat point or a downbeat point is called a beat point.
Definition 5.2.
Let be an Alexandroff space (with the distinguished point ). A retraction () is called:

a comparative retraction, if for every ,

an upretraction, if for every ,

a downretraction, if for every ,

a retraction removing a beat point, if exists an being an upbeat point under some or a downbeat point over some and such that or , and for all .
Remark 5.3.
Retractions removing a beat point, up and downretractions are all comparative. Moreover, every comparative retraction may be written as a composition of an upretraction and a downretraction: if is a comparative retraction, then , where
and
It is straightforward to check and are orderpreserving.
Moreover, by Lemma 4.13, , where , is homotopic to by a homotopy keeping fixed the set , and this means is a strong deformation retraction.
Definition 5.4.
In the category of Alexandroff spaces (with distinguished points) let denote the class of all comparative retractions, and classes of, respectively, up and downretractions and the class of retractions removing a beat point.
Definition 5.5.
Let be a class of retractions in the category of Alexandroff spaces (with distinguished points). A nonempty Alexandroff space () is called an core if there is no retraction () in other than identity.
By Remark 5.3, a space being a core is equivalent to that space being an core. Also, every core is an core. In general, the reverse implication does not hold, as we can see in the following example.
Example 5.6.
Let denote the rational numbers with the standard order. is an core. On the other hand, is a comparative retraction.
For another example, consider the space with order given by: iff or . is an core. But consider the retractions given by ( indicates the standard addition of natural numbers), for all . We have: , so is not a core. Moreover, , so is contractible.
Since in this article we are mostly concerned with fpspaces and locally finite spaces, the following proposition guarantees that in our setting cores and cores are the same. The proposition was proved in [19] for finite spaces.
Proposition 5.7.
Let be a finitechains core (with the distinguished point ). If for some , then . Therefore, a finitechains core is a core.
Proof.
Let (or ) be a continuous map. Suppose . We will show .
Since satisfies DCC and Lemma 3.8 holds, we may use Noetherian induction. Take an and assume that for all . By continuity of , for . But , so . If , then must be a downbeat point over . Otherwise, an would exist with and would not be orderpreserving. But is an core, so it contains no beat points. Therefore, . By induction, .
If , then by the same reasoning as above. Therefore, is an core and thus a core. ∎
In [19] Stong shows that for any finite space it is possible to reduce it to its core by removing, one by one, beat points. Since the used retractions are comparative and Lemma 4.13 holds, the resulting core is a strong deformation retract of the given finite space (and thus the space and its core are homotopy equivalent). Moreover, because of Proposition 5.7, any function comparable to is equal to . But is finite, so is Alexandroff and that means is an isolated point in . In particular, any homotopic to is equal to . Now it easily follows (like in the proof of Corollary 5.17) that any two finite spaces are homotopy equivalent if and only if their cores are homeomorphic. (Using the same arguments one can show that finite spaces with distinguished points are homotopy equivalent if and only if their cores are homeomorphic.)
From [8] we cite (in a version modified to our needs) the LiMilner theorem, which allows us to generalise the above results to chaincomplete posets without infinite antichains.
Theorem 5.8 (cf. Theorem 6.11 of [8]).
For every chaincomplete partially ordered set with no infinite antichains exists a finite core that is a strong deformation retract of . (In fact, is dismantlable to , in the sense of Definition 5.9, in finitely many steps.)
Therefore, any two chaincomplete ordered sets without infinite antichains are homotopy equivalent iff their cores (which exist and are finite) are homeomorphic.
Question 2.
Is there a version of the above theorem for posets with distinguished points?
Now we will follow Stong’s path in the setting of fpspaces and bpspaces. We shall describe a process which we call cdismantling. It is a special case of the following, more general definition of dismantling a poset.
Definition 5.9 (cf. Exercise 24 in Chapter 4 of [18]).
Let be an ordinal and be an Alexandroff space (with the distinguished point ). Let be a family of retractions from a class (keeping the distinguished point fixed) such that , for all and for limit ordinals . By transfinite induction we define a family of retractions :

,

,

for a limit ordinal and an , if there exists such that for all , then .
The induction ends when is defined or when cannot be fully defined for some limit ordinal . In the first case we say the family is infinitely composable and is dismantlable to (in steps). In the second case we say the family is not infinitely composable and if the induction stopped at and , we say is dismantlable to the empty set (in steps).
Remark 5.10.
By the standard construction of categorical limits, it is clear that the set together with the family of inclusions is the limit of the inverse system of inclusions.
Suppose the sequence is infinitely composable. We show together with the family of retractions is also the colimit of the direct system of retractions induced in the obvious way by the sequence .
It is trivial that for all . If