Computable Topological Space Research Articles

Computability of Generalized Graphs

We investigate conditions under which a semicomputable set is computable. In particular, we study topological pairs (A,B) which have a computable type, which means that in any computable topological space, a semicomputable set S is computable if there exists a semicomputable set T such that (S,T) is homeomorphic to (A,B). It is known that (G,E) has a computable type if G is a topological graph and E is the set of all its endpoints. Furthermore, the same holds if G is a so-called chainable graph. We generalize the notion of a chainable graph and prove that the same result holds for a larger class of spaces.

Separating notions in effective topology

We compare several natural notions of effective presentability of a topological space up to homeomorphism. We note that every left-c.e. (lower-semicomputable) Stone space is homeomorphic to a computable one. In contrast, we produce an example of a locally compact, left-c.e. space that is not homeomorphic to any computable Polish space. We apply a similar technique to produce examples of computable topological spaces not homeomorphic to any right-c.e. (upper-semicomputable) Polish space, and indeed to any arithmetical or even analytical Polish space. We then apply our techniques to totally disconnected locally compact (tdlc) groups. We prove that every computably locally compact tdlc group is topologically isomorphic to a computable tdlc group.

Computable Stone spaces

We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a computable topological Polish space not homeomorphic to any computably metrized space. In fact, in our proof we construct a right-c.e. metrized Stone space which is not homeomorphic to any computably metrized space. Then we introduce a new notion of effective categoricity for effectively compact spaces and prove that effectively categorical Stone spaces are exactly the duals of computably categorical Boolean algebras. Finally, we prove that, for a Stone space X, the Banach space C(X;R) has a computable presentation if, and only if, X is homeomorphic to a computably metrized space. This gives an unexpected positive partial answer to a question recently posed by McNicholl.

Computability of glued manifolds

Abstract We examine conditions under which a semicomputable set in a computable topological space is computable. In particular, we examine topological spaces $\varDelta $ that have computable type, which means that any semicomputable set homeomorphic to $\varDelta $ is computable. It is known that each compact manifold has computable type. In this paper, we examine compact manifolds $M$ and $N$ and a space $M\cup _{\gamma }N$ obtained by gluing $M$ and $N$ together by way of a homeomorphism $\gamma :A\rightarrow B$, where $A$ and $B$ are closed subspaces of $M$ and $N$, respectively. We show that $M\cup _{\gamma }N$ in general need not have computable type. We prove that $M\cup _{\gamma }N$ has computable type under the additional assumption that $A$ and $B$ are contained in regular submanifolds of $M$ and $N$. We also show that the same holds for a space obtained by gluing finitely many manifolds, but not for infinitely many.

Computable subcontinua of semicomputable chainable Hausdorff continua

In the setting of computable topological spaces, we consider semicomputable, chainable Hausdorff continua. If such a continuum S is decomposable, it can be computably approximated; more precisely, for every open cover U, there exist two computable points aˆ, bˆ and a computable subcontinuum Sˆ that is chainable from aˆ to bˆ and is U-close to S.

Computability of Products of Chainable Continua

We examine conditions under which a semicomputable set in a computable topological space is computable. In particular, we examine topological pairs (A, B) with the following property: if X is a computable topological space and $f:A\rightarrow X$ is an embedding such that f(A) and f(B) are semicomputable sets in X, then f(A) is a computable set in X. Such pairs (A, B) are said to have computable type. It is known that $(\mathcal {K},\{a,b\})$ has computable type if $\mathcal {K}$ is a Hausdorff continuum chainable from a to b. It is also known that (In, ∂In) has computable type, where In is the n-dimensional unit cube and ∂In is its boundary in $\mathbb {R}^{n} $ . We generalize these results by proving the following: if $\mathcal {K}_{i} $ is a nontrivial Hausdorff continuum chainable from ai to bi for $i\in \{1,{\dots } ,n\}$ , then $({\prod }_{i=1}^{n} \mathcal {K}_{i} ,B)$ has computable type, where B is the set of all $(x_{1} ,{\dots } ,x_{n})\in {\prod }_{i=1}^{n} \mathcal {K}_{i}$ such that xi ∈{ai, bi} for some $i\in \{1,{\dots } ,n\}$ .

Computability of pseudo-cubes

We examine topological pairs (Δ,Σ) which have computable type: if X is a computable topological space and f:Δ→X a topological embedding such that f(Δ) and f(Σ) are semicomputable sets in X, then f(Δ) is a computable set in X. It is known that (D,W) has computable type, where D is the Warsaw disc and W is the Warsaw circle. In this paper we identify a class of topological pairs which are similar to (D,W) and have computable type: we prove that (Δ,Σ) has computable type if Δ can somehow be approximated by a 2-cell whose boundary circle approximates Σ. Moreover, we examine higher-dimensional analogues of such pairs.

Computability of graphs

AbstractWe consider topological pairs , , which have computable type, which means that they have the following property: if X is a computable topological space and a topological imbedding such that and are semicomputable sets in X, then is a computable set in X. It is known, e.g., that has computable type if M is a compact manifold with boundary. In this paper we examine topological spaces called graphs and we show that we can in a natural way associate to each graph G a discrete subspace E so that has computable type. Furthermore, we use this result to conclude that certain noncompact semicomputable graphs in computable metric spaces are computable.

Chainable and circularly chainable semicomputable sets in computable topological spaces

We examine conditions under which, in a computable topological space, a semicomputable set is computable. It is known that in a computable metric space a semicomputable set S is computable if S is a continuum chainable from a to b, where a and b are computable points, or S is a circularly chainable continuum which is not chainable. We prove that this result holds in any computable topological space.

Semicomputable manifolds in computable topological spaces

We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable in ambient spaces which are second countable Hausdorff. We prove that a semicomputable compact manifold M is computable if its boundary ∂M is computable. We also show how this result combined with a certain construction which compactifies a semicomputable set leads to the conclusion that some noncompact semicomputable manifolds in computable metric spaces are computable.

Computability on measurable functions

For a computable measure space with computable σ -finite measure we study computability on the space of measurable functions. First we prove that for a natural multi-representation the measurable sets are closed under finite union and intersection and under countable union. We introduce a natural multi-representation of the measurable functions to a computable topological space and prove that composition with continuous functions is computable. Canonically, for a computable metric space the associated computable topological space has a basis of balls B(a, s) with center from the dense set and rational radius. From a given sequence (μk)k of finite Borel measures we can compute a dense sequence (rj )j of radii such that μk(S) = 0f o ra l lk and all spheres S = S(a,rj ) with a from the dense set and j ∈ N. For measurable functions to computable compact computable metric spaces the natural multi-representation is equivalent to a multi-representation via fast uniformly converging sequences of simple functions. For non-negative measurable numerical functions the sequences can be chosen to be non-decreasing. Some results on integration are added.

Algorithmic randomness over general spaces

The study of Martin‐Löf randomness on a computable metric space with a computable measure has seen much progress recently. In this paper we study Martin‐Löf randomness on a more general space, that is, a computable topological space with a computable measure. On such a space, Martin‐Löf randomness may not be a natural notion because there is no universal test, and Martin‐Löf randomness and complexity randomness (defined in this paper) do not coincide in general. We show that SCT3 is a sufficient condition for the existence and coincidence, and study how much we can weaken this condition.

Computably regular topological spaces

This article continues the study of computable elementary topology started by the author and T. Grubba in 2009 and extends the author's 2010 study of axioms of computable separation. Several computable T3- and Tychonoff separation axioms are introduced and their logical relation is investigated. A number of implications between these axioms are proved and several implications are excluded by counter examples, however, many questions have not yet been answered. Known results on computable metrization of T3-spaces from M. Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are proved under uniform assumptions and with partly simpler proofs, in particular, the theorem that every computably regular computable topological space with non-empty base elements can be embedded into a computable metric space. Most of the computable separation axioms remain true for finite products of spaces.

Computability of measurable sets via effective topologies

We investigate in the frame of TTE the computability of functions of the measurable sets from an infinite computable measure space such as the measure and the four kinds of set operations. We first present a series of undecidability and incomputability results about measurable sets. Then we construct several examples of computable topological spaces from the abstract infinite computable measure space, and analyze the computability of the considered functions via respectively each of the standard representations of the computable topological spaces constructed.