Co-c.e. spheres and cells in computable metric spaces

We investigate conditions under which a co-computably enumerable set in a computable metric space is computable. Using higher-dimensional chains and spherical chains we prove that in each computable metric space which is locally computable each co-computably enumerable sphere is computable and each co-c.e. cell with co-c.e. boundary sphere is computable

Compact manifolds with computable boundaries

We investigate conditions under which a co-computably enumerable closed set in a computable metric space is computable and prove that in each locally computable computable metric space each co-computably enumerable compact manifold with computable boundary is computable. In fact, we examine the notion of a semi-computable compact set and we prove a more general result: in any computable metric space each semi-computable compact manifold with computable boundary is computable. In particular, each semi-computable compact (boundaryless) manifold is computable

On the topological aspects of the theory of represented spaces

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escard\'o, Schr\"oder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and -- closely linked to these -- properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.Comment: Earlier versions were titled "Compactness and separation for represented spaces" and "A new introduction to the theory of represented spaces

Semicomputable Geometry

Computability and semicomputability of compact subsets of the Euclidean spaces are important notions, that have been investigated for many classes of sets including fractals (Julia sets, Mandelbrot set) and objects with geometrical or topological constraints (embedding of a sphere). In this paper we investigate one of the simplest classes, namely the filled triangles in the plane. We study the properties of the parameters of semicomputable triangles, such as the coordinates of their vertices. This problem is surprisingly rich. We introduce and develop a notion of semicomputability of points of the plane which is a generalization in dimension 2 of the left-c.e. and right-c.e. numbers. We relate this notion to Solovay reducibility. We show that semicomputable triangles admit no finite parametrization, for some notion of parametrization

Computability of 1-manifolds

A semi-computable set S in a computable metric space need not be computable. However, in some cases, if S has certain topological properties, we can conclude that S is computable. It is known that if a semi-computable set S is a compact manifold with boundary, then the computability of \deltaS implies the computability of S. In this paper we examine the case when S is a 1-manifold with boundary, not necessarily compact. We show that a similar result holds in this case under assumption that S has finitely many components

Strong computable type

A compact set has computable type if any homeomorphic copy of the set which is semicomputable is actually computable. Miller proved that finite-dimensional spheres have computable type, Iljazović and other authors established the property for many other sets, such as manifolds. In this article we propose a theoretical study of the notion of computable type, in order to improve our general understanding of this notion and to provide tools to prove or disprove this property. We first show that the definitions of computable type that were distinguished in the literature, involving metric spaces and Hausdorff spaces respectively, are actually equivalent. We argue that the stronger, relativized version of computable type, is better behaved and prone to topological analysis. We obtain characterizations of strong computable type, related to the descriptive complexity of topological invariants, as well as purely topological criteria. We study two families of topological invariants of low descriptive complexity, expressing the extensibility and the nullhomotopy of continuous functions. We apply the theory to revisit previous results and obtain new ones

Aspects Topologiques des Représentations en Analyse Calculable

Computable analysis provides a formalization of algorithmic computations over infinite mathematical objects. The central notion of this theory is the symbolic representation of objects, which determines the computation power of the machine, and has a direct impact on the difficulty to solve any given problem. The friction between the discrete nature of computations and the continuous nature of mathematical objects is captured by topology, which expresses the idea of finite approximations of infinite objects.We thoroughly study the multiple interactions between computations and topology, analysing the information that can be algorithmically extracted from a representation. In particular, we focus on the comparison between two representations of a single family of objects, on the precise relationship between algorithmic and topological complexity of problems, and on the relationship between finite and infinite representations.L’analyse calculable permet de formaliser le traitement algorithmique d’objets mathématiques infinis. La théorie repose sur une représentation symbolique des objets, dont le choix détermine les capacités de calcul de la machine, notamment sa difficulté à résoudre chaque problème donné. La friction entre le caractère discret du calcul et la nature continue des objets est capturée par la topologie, qui exprime l’idée d’approximation finie d’objets infinis.Nous étudions en profondeur les multiples interactions entre calcul et topologie, cherchant à analyser l’information qui peut être extraite algorithmiquement d’une représentation. Je me penche plus particulièrement sur la comparaison entre deux représentations d’une même famille d’objets, sur les liens détaillés entre complexité algorithmique et topologique des problèmes, ainsi que sur les relations entre représentations finies et infinies

Computability of sets with disconnected complements

U ovom doktorskom radu proučavali su se uvjeti koji omogućuju da skupovi s nepovezanim komplementom sadrže izračunljivu točku, odnosno postaju izračunljivi. Štoviše, istraživanje ove disertacije obuhvaća i skupove koji su homeomorfni skupovima s nepovezanim komplementima. Ambijentni prostor nad kojim su dokazivani rezultati ovog rada bio je izračunljiv metrički prostor $(X, d, \alpha)$, koji predstavlja svojevrsno poopćenje standardnog izračunljivog euklidskog prostora $\mathbb{R}^n$. U poglavlju 1. definirali su se neki osnovni pojmovi iz izračunljive analize te su istaknuti neki poznati, ali za ovo istraživanje, važni rezultati, koji su se tijekom ove doktorske disertacije koristili. Naglasak ovog poglavlja je posebno bio na doprinosima ove doktorske disertacije. Cilj ovog rada među ostalim je također bio da se još jednom istakne kako postoji uska veza između topologije i same teorije izračunljivosti. Neka topološka svojstva mogu učiniti poluizračunljiv/co-c.e. skup izračunljivim ili barem učiniti da takvi skupovi sadrže izračunljivu točku. Originalni doprinos ovog rada započet je s poglavljem 2 u kojem su proučavani co-c.e. skupovi s nepovezanim komplementima u izračunljivom metričkom prostoru. Pažnja se posebno usmjerila na slučaj kada je izračunljiv metrički prostor $(X, d, \alpha)$ efektivno lokalno povezan i kada se komponente povezanosti komplementa co-c.e. skupa $S \subseteq X$ mogu na efektivan način razlikovati. Opisali su se neki dovoljni uvjeti koji omogućavaju da takav skup $S$ sadrži izračunljivu točku te neki dovoljni uvjeti uz koje skup $S$ postaje izračunljiv. Slobodno se može istaknuti da je pojam efektivne lokalne povezanosti zapravo ključ proučavanja ovog poglavlja. Stoga je u potpoglavlju 2.2. dana precizna definicija toga pojma. Budući se definicija efektivne lokalne povezanosti ne treba ograničiti samo na povezane otvorene kugle stvara se potreba za opisom pojma koji bi objašnjavao značenje izračunljivosti niza otvorenih (povezanih) skupova $U_i$ u izračunljivom metričkom prostoru $(X, d, \alpha)$. U tom kontekstu prezentira se definicija koja objašnjava značenje da niz skupova $(A_i)$ efektivno profinjuje niz skupova $(B_i)$ te značenje kada su dva niza skupova $(A_i)$ i $(B_i)$ izračunljivo ekvivalentni. U potpoglavlju 2.4. uvodi se pojam efektivne nepovezanosti otvorenog nepovezanog skupa $V$ u $(X, d, \alpha)$. Tim pojmom poopćava se situacija koja se odnosi na slučaj kada nepovezan otvoren skup $V$ u $(X, d, \alpha)$ ima konačno mnogo komponenti povezanosti. Glavni rezultat ovog poglavlja, čiji je dokaz dan u potpoglavlju 2.4., kaže da ukoliko je $(X, d, \alpha)$ efektivno lokalno povezan izračunljiv metrički prostor te ukoliko je $S$ co-c.e. skup u $(X, d, \alpha)$ takav da je $X \setminus S$ efektivno nepovezan i takav da svaka točka skupa $S$ leži na rubu barem dvije komponente povezanosti od $X \setminus S$, onda skup $S$ nužno mora biti izračunljiv u $(X, d, \alpha)$ (vidi teorem 2.4.5 u potpoglavlju 2.4.). Također je dokazano da u efektivno lokalno povezanom izračunljivom metričkom prostoru $(X, d, \alpha)$, proizvoljan co-c.e. skup $S$, sa svojstvom da je $X \setminus S$ efektivno nepovezan, nužno sadrži izračunljivu točku ukoliko se dodatno pretpostavi da je metrički prostor $(X, d)$ povezan i potpun (korolar 2.4.6). Na kraju tog poglavlja prezentiran je još jedan dovoljan uvjet uz koji co-c.e. skup s efektivno nepovezanim komplementom u potpunom, efektivno lokalno povezanom izračunljivom metričkom prostoru $(X, d, \alpha)$ sadrži izračunljivu točku. Skup S sadrži izračunljivu točku $x$ ako postoji neki povezan skup $A$ koji siječe dvije različite komponente povezanosti skupa $X \setminus S$. Štoviše, takav se $x$ može naći po volji blizu skupa $A$ (vidi teorem 2.4.8). Poglavlje 3. je prije svega posvećeno poopćenju takozvane izračunljive verzije Bolzanovog teorema o nultočki. Ukoliko je skup $K$ izračunljiv i povezan podskup od $\mathbb{R}^2$ koji siječe obje komponente skupa $\mathbb{R}^2 \setminus S$, gdje je $S = \mathbb{R} \times \{0\}$, onda $K$ nužno siječe $S$ jer je $K$ povezan. Postavlja se stoga sljedeće pitanje: mora li nužno $K$ sijeći skup $S$ u izračunljivoj točki? Izračunljiva verzija Bolzanovog teorema o nultočki nam kaže da ukoliko za skup $K$ uzmemo graf izračunljive funkcije $f : [0; 1] \to \mathbb{R}$, takve da je $f(0) 0$, onda $K$ mora sijeći $S$ u izračunljivoj točki. U potpoglavlju 3.3. ovaj rezultat se poopćava na način koji ćemo sada ukratko opisati. Prvo se promatra izračunljiv metrički prostor $(X, d, \alpha)$ i dva c.e. otvorena skupa $U$ i $V$ u $X$. Skup $S$ se definira kao komplement skupa $U \cup V$ u $X$. Sada se pretpostavi da je $K$ kontinuum u $X$ koji je lančast od točke $a$ do točke $b$, gdje je $a \in U$ i $b \in V$ , te se pretpostavi da je $K$ izračunljiv skup i da su točke $a$ i $b$ izračunljive. Uz dodatnu pretpostavku da je $K \cap S$ potpuno nepovezan pokazuje se da skup $K \cap S$ sadrži izračunljivu točku (vidi teorem 3.3.3). Ideja koja se je nalazila u pozadini dokaza ovog teorema je bila da se na izvjestan način uspije skup $K$ prekriti sa $(U, V )$-lancem od točke $a$ do točke $b$, to jest lancem koji ima svojstvo da za svake dvije susjedne karike barem jedna leži u skupu $U$ ili u skupu $V$ , te da prva karika ovakvog lanca sadrži točku $a$, a posljednja karika sadrži točku $b$ (vidi potpoglavlje 3.1.). Iako se pretpostavka potpune nepovezanosti skupa $K \cap S$ može na prvi pogled učiniti dosta neobičnom i stranom, ona je usko povezana sa samom tehnikom $(U, V)$-lanaca. Također, za potrebe dokaza samog teorema 3.3.3 razvija se koncept separatora, augmentatora i lokatora (potpoglavlje 3.2.) da bi se dobila mogućnost efektivnog načina lociranja tražene izračunljive točke u skupu $K \cap S$. Kao posljedica teorema 3.3.3, dobiva se rezultat koji se odnosi na luk. Ukoliko se promatra izračunljiv metrički prostor $(X, d, \alpha)$ i dva c.e. otvorena skupa $U$ i $V$ u $X$, te ukoliko je $S := X \setminus (U \cup V )$, tada svaki izračunljiv luk $A$ u $X$, s krajnjim izračunljivim točkama $a \in U$ i $b \in V$ , nužno mora sijeći skup $S$ u izračunljivoj točki (vidi teorem 3.3.4). Ovdje ćemo još jednom naglasiti da u ovom prethodno navedenom teoremu pretpostavka da je skup $A \cap S$ potpuno nepovezan iščezava, a ipak se navedeni rezultat dokazuje kao posljedica teorema 3.3.3 u kojem je pretpostavka potpune nepovezanosti neizostavna. U potpoglavlju 3.4. prethodna dva rezultata se poopćavaju na način da se pretpostavi samo da skupovi $K$ i $A$ sijeku skupove $U$ i $V$ (izostavili smo pretpostavku koja se je odnosila na važnost točaka $a \in U$ i $b \in V$). Detalje navedenog poopćenja čitatelj može pratiti kroz teorem 3.4.14. Taj teorem istaknimo kao najveći doprinos ovog poglavlja. Također, u potpoglavlju 3.4. promatra se i slučaj kada $K \cap S$ nije potpuno nepovezan. Prvo se u ovom potpoglavlju može uočiti da ukoliko je $K$ izračunljiv lančasti kontinuum, te $S$ proizvoljan co-c.e. zatvoren skup, i ukoliko je promatrani skup $K \cap S$ povezan, onda je nužno $K \cap S$ poluizračunljiv lančasti kontinuum (to je zapravo posljedica propozicije 3.4.2). Stoga je temeljno pitanje koje prožima ovo potpoglavlje 3.4. bilo: što se općenito može kazati o izračunljivim točkama i izračunljivosti poluizračunljivih lančastih kontinuuma? U teoremu 3.4.7 promatra se izračunljiv metrički prostor $(X, d, \alpha)$ te poluizračunljiv skup $S$ u ovom prostoru. Ukoliko se pretpostavi da je $S$ kao potprostor od $(X, d$, lančasti kontinuum, te se dodatno pretpostavi da postoje dva potkontinuuma $K_1$ i $K_2$ od $S$ takva da je $S = K_1 \cup K_2$, te da postoje neke dvije točke $a$ i $b$ sa svojstvom da je $a \in K_1 \setminus K_2$ i $b \in K_2 \setminus K_1$, onda se dokazalo da skup $S$ sadrži izračunljivu točku. Štoviše, u tom se teoremu dobiva da su izračunljive točke zapravo guste u skupu $S$. Kao direktnu posljedicu toga teorema dobiva se sličan rezultat i za luk (vidi korolar 3.4.8). Kroz teorem 3.4.9, koji teorem 3.4.7 opisuje u terminu pojma dekompozabilnosti, uvidjeli smo da svaki dekompozabilan poluizračunljiv lančasti kontinuum $S$ u izračunljivom metričkom prostoru $(X, d, \alpha)$ mora sadržavati izračunljivu točku. Nadalje, važan rezultat isprofilirao se je i kroz teorem 3.4.10 koji kaže da u izračunljivom metričkom prostoru $(X, d, \alpha)$, svaki poluizračunljiv lančasti kontinuum, koji ima izoliranu i dekompozabilnu komponentu povezanosti nužno mora sadržavati izračunljivu točku (štoviše izračunljive točke su guste u toj komponenti povezanosti). Budući se kroz ovo potpoglavlje također htjelo i poopćiti rezultat koji kaže da je svaka poluizračunljiva topološka kružnica $S$ u izračunljivom metričkom prostoru $(X, d, \alpha)$ izračunljiva, dokazujemo i teorem 3.4.12. U tom teoremu je dokazano da je proizvoljan poluizračunljiv cirkularno lančasti kontinuum $S$ u $(X, d, \alpha)$, koji nije lančast, nužno izračunljiv skup (ovdje, u ovoj disertaciji, ovaj rezultat je dokazan bez korištenja pretpostavke da je $(X, d, \alpha)$ lokalno izračunljiv). Na kraju samog poglavlja 3. pred čitatelja se stavljaju neka otvorena pitanja koja su se prirodno pojavila pri istraživanju navedenih problema ovog poglavlja. U posljednjem poglavlju ove disertacije, poglavlju 4., nastavljaju se izučavati uvjeti koji bi omogućili da poluizračunljiv skup u izračunljivom metričkom prostoru postane izračunljiv. Topologija igra veoma važnu ulogu pri opisu takvih uvjeta, kao što se do sada moglo uočiti. Glavna motivacija za ovo poglavlje bio je dobro poznati rezultat koji kaže da je poluizračunljiva ćelija izračunljiva uz dodatnu pretpostavku da je njezina rubna sfera izračunljiv skup. U tom kontekstu, kroz cijelo poglavlje 4., proučava se poluizračunljivi varšavski disk i njegova rubna varšavska kružnica. U teoremu 4.4.1 dokazuje se da je poluizračunljiv varšavski disk u izračunljivom metričkom prostoru $(X, d, \alpha)$ izračunljiv ukoliko je njegova rubna varšavska kružnica poluizračunljiv skup. Glavna ideja dokaza ovog teorema sastojala se je u tome da se nekako aproksimira varšavski disk s 2-ćelijama (vidi potpoglavlje 4.2.) te da se onda nekako iskoristi tehnika 2-lanaca (vidi posebno propoziciju 4.2.1 te lemu 4.2.3). Važnost glavnog rezultat ovog poglavlja (zajedno s tehnikama koje su razvijene u svrhu njegova dokaza) jest u tome što bi on trebao pridonijeti još boljem razumijevanju veze koja se neminovno pojavljuje između topologije i same izračunljivosti, te naposlijetku svakako treba istaknuti da ovaj rezultat pridonosi dubljem shvaćanju važnosti takozvanog „rubnog uvjeta“.In this dissertation we have examined under what conditions sets with disconnected complements contain a computable point, or they become computable. Moreover, we have generalized our research also on sets which are homeomorphic to those sets with disconnected complements. The ambient space for this research was computable metric space $(X, d, \alpha)$, which represents a kind of generalization of standard computable Euclidean space $\mathbb{R}^n$.We can say that a computable metric space is a metric space in which we impose computability notions using some fixed dense sequence $\alpha)$. In chapter 1. we defined some basic concepts of computable analysis and we referred readers to some important results which we used throughout the whole dissertation. The main part of this chapter 1. was to emphasize the contribution of our research. We strongly believe that this paper has once more showed to us that there is a strong connection between topology and computability. Some topological properties can force a semicomputable/co-c.e. sets to become computable or at least to have computable points We started our main work with chapter 2 in which we have been examining co-c.e. sets with disconnected complements in a computable metric space. We focused on the case when the computable metric space $(X, d, \alpha)$ is effectively locally connected and when the connected components of the complement of a co-c.e. set $S \subseteq X$ can be effectively distinguished. We gave a sufficient condition that such an $S$ contains a computable point and a sufficient condition that $S$ is computable. We can say that the notion of effective local connectedness is the key in this chapter. We gave precise definition of that term. Because we did not want to restrict ourselves to connected open balls, we introduced in section 2.2. a certain notion that a sequence of open (connected) sets $U_i$ is computable in $(X, d, \alpha)$. To do this we needed to define when some sequence of sets $(A_i)$ effectively refines sequence of sets $(B_i)$, and we needed a concept that two sequences of sets $(A_i)$ and $(B_i)$ are computably equivalent. Also in section 2.4. we introduced the concept which describes when some disconnected open set $V$ in $(X, d, \alpha)$ is effectivelly disconnected because we wanted to have a notion which will generalize the situation when disconnected open set $V$ in $(X, d, \alpha)$ has finitely many components. The main result that we have proved in chapter 2. says that if $(X, d, \alpha)$ is an effectively locally connected computable metric space and if $S$ is a co-c.e. set in $(X, d, \alpha)$ such that $X \setminus S$ is effectivelly disconected and such that each point of $S$ lies in the boundary of at least two different components of $X \setminus S$, then $S$ needs to be a computable set in $(X, d, \alpha)$ (see theorem 2.4.5 in section 2.4.). We have also proved that in effectively locally connected computable metric space $(X, d, \alpha)$, any co-c.e. set $S$, such that $X \setminus S$ is effectively disconnected, has to contain a computable point if we additionaly assume that metric space $(X, d)$ is connected and complete (see corollary 2.4.6). We gave another sufficient condition that co-c.e. set with effectively disconnected complement in complete effectively locally connected computable metric space $(X, d, \alpha)$ contains a computable point. We showed that $S$ contains a computable point $x$ if there exists a connected set $A$ which intersects two different components of $X \setminus S$. Moreover, we proved that such $x$ can be found sufficiently close to $A$ (see theorem 2.4.8). Chapter 3 was devoted to generalization of computable version of Bolzano’s theorem. If we assume that $K$ is some computable and connected subset of Euclidean plane $\mathbb{R}^2$ which intersects both components of $\mathbb{R}^2 \setminus S$, where $S = \mathbb{R} \times \{0\}$, then $K$ certainly intersects $S$ because $K$ is connected. The question is does $K$ needs to intersect $S$ in a computable point? The computable version of Bolzano’s theorem states that if we take for a set $K$ to be a graph of a computable function $f : [0; 1] \to \mathbb{R}$, such that $f(0) 0$, then $K$ intersects $S$ in a computable point. In section 3.3. we have generalized this theorem in the following way. First, we observed computable metric space $(X, d, \alpha)$ and two c.e. open sets $U$ and $V$ in $X$. We defined set $S$ to be the complement of the set $U \cup V$ in $X$. Now, we assumed that $K$ is a continuum in $X$ chainable from $a$ to $b$, where a $a \in U$ and $b \in V$ , and we assumed that $K$ is a computable set and that points $a$ and $b$ are computable. Additionally, we assumed that $K \cap S$ is totally disconnected. Then we proved that $K \cap S$ contains a computable point (see theorem 3.3.3). The idea of the proof of that theorem was to cover set $K$ by $(U, V )$-chains, by chains which have the property that for each two adjacent links at least one lies in $U$ or $V$ (see section 3.1.). The necessity of the assumption of total disconnectedness of $K \cap S$ is closely related to the technique of $(U, V )$-chains. Also to prove the theorem 3.3.3 we needed to develop some techniques of separators, augmentators and locators (section 3.2.) to be able to precisely (effectively) locate the required computable point in the set $K \cap S$. As a consequence of the theorem 3.3.3, we have gotten the result for an arc. If we observe a computable metric space $(X, d, \alpha)$ and two c.e. open sets $U$ and $V$ in $X$, and if we define $S := X \setminus (U \cup V )$, then each computable arc $A$ in $X$, with computable endpoints $a \in U$ and $b \in V$ needs to intersect the set $S$ in a computable point (see theorem 3.3.4). We have to emphasize again that in this theorem we did not need to assume that the set $A \cap S$ is totally disconnected, and yet we have proved this theorem using the theorem 3.3.3. In section 3.4. we have generalized these two results assuming that this two sets $K$ and $A$ intersect sets $U$ and $V$ (we dropped out the assumption which referred to importance of points $a \in U$ and $b \in V$ ). The details of this generalization, reader can observe in theorem 3.4.14. Also in section 3.4. we have examined the case when $K \cap S$is not totally disconnected. First in this section we saw that if $K$ is a computable chainable continuum and $S$ is any co-c.e. closed set and if $K \cap S$ is connected, then $K \cap S$ is a semi-computable chainable continuum (actually this is a consequence of the proposition 3.4.2). So the general question, which we have dealt with in section 3.4. was: what can be in general said about computable points and computability of semi-computable chainable continua? In theorem 3.4.7 we observed computable metric space $(X, d, \alpha)$ and a semi-computable set $S$ in that space. We assumed that $S$ is, as a subspace of $(X, d$, chainable continuum and additionally we assumed that there exist two subcontinua $K_1$ and $K_2$ of $S$ such that $S = K_1 \cup K_2$ and that there exist two points $a$ and $b$ such that $a \in K_1 \setminus K_2$ and $b \in K_2 \setminus K_1$. Then we proved that $S$ contains a computable point. Moreover, in that theorem we have proved that computable points are dense in $S$. As a direct consequence of the theorem 3.4.7 we got the same result for an arc (see corollary 3.4.8). Throughout the theorem 3.4.9 we saw that any decomposable semi-computable chainable continuum $S$ in computable metric space $(X, d, \alpha)$ has to contain a computable point (this theorem is closely related to the theorem 3.4.7 and it puts that theorem in the context of decomposability). Also, an important result came out through theorem 3.4.10 which claims that in computable metric space $(X, d, \alpha)$, any semi-computable chainable continuum, which has isolated and decomposable connected component, has a computable point, moreover computable points are dense in that connected component. Further, we wanted to generalize a result which says that any semi-computable topological circle $S$ in computable metric space $(X, d, \alpha)$ has to be a computable set. We did it throughout the theorem 3.4.12. In that theorem we have proved that any semi-computable circulary chainable continuum $S$ in $(X, d, \alpha)$, which is not chainable, needs to be a computable set (and we did it without assumption that $(X, d, \alpha)$ is locally computable). As a conclusion of the chapter 3. we gave some open questions which naturally appeared while we were working on this problems. Last chapter of this dissertation, chapter 4., was dedicated to finding out more some conditions which will allow a semi-computable set in computable metric space to became computable. We know that topology plays an important role in the description of such conditions. The main motivation for this chapter was the well known result which says that a semi-computable cell is computable if its boundary sphere is computable. In that context, throughout the chapter 4, we have been investigating semi-computable Warsaw discs and their boundary Warsaw circles. We proved in theorem 4.4.1 that a semicomputable Warsaw disc in computable metric space $(X, d, \alpha)$ is computable if its boundary Warsaw circle is semi-computable. The main idea behind the proof of this theorem was to approximate the Warsaw disc by 2-cells (see section 4.2.) and then to adjust somehow the technique of 2-chains (see especially proposition 4.2.1 and lemma 4.2.3). We believe that the main result of this chapter and the techniques that we used in its proof contribute to the better understanding of relationship between topology and computability and also, in particular, to the understanding of the so-called “boundary condition”