Naslov
Computable Type of Certain Quotient Spaces

Autori
Čelar, Matea ; Iljazović, Zvonko

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
World Logic Day 2023 Zagreb - Book of Abstracts / - , 2023, 4-4

Skup
World Logic Day 2023 Zagreb

Mjesto i datum
Zagreb, Hrvatska, 13.01.2023. - 14.01.2023

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Nije recenziran

Ključne riječi
Computable Type of Certain Quotient Spaces

Sažetak
A closed set S in a computable topological space is said to be computably enumerable if it is possible to semidecide whether a basic open set intersects S. Furthermore, a compact set in a computable topological space is said to be semicomputable if it is possible to semidecide whether a finite union of basic open sets covers S. A set which is both computably enumerable and semicomputable is called computable. Topology plays an important role in determining the relationship between semicomputability and computable enumerability. In particular, semicomputable sets with certain topological properties are necessarily computably enumarable (and, therefore, computable). This is expressed in the notion of computable type: a topological space A is said to have computable type if every semicomputable set homeomorphic to A must be computable. More generally, topological pair (A, B) has computable type if, whenever A is embedded in a computable topo- logical space, semicomputability of images of A and B implies that the image of A is computable. Some known examples of spaces with computable type are topological manifolds, chainable and circularly chainable continua and finite graphs ([3, 2, 4]). It is known that both the pair (Bn, Sn−1) of the unit ball and its boundary and the quotient space Bn/Sn−1 ∼= Sn have computable type ([1]). Motivated by this, we consider the effect of quotients on preserving computable type. We prove the following: Theorem 1 Let A be a topological space and let B be a compact subset of A such that IntA B = ∅. If A/B has computable type, then (A, B) has computable type. However, if (A, B) has computable type, A/B generally need not have com- putable type even if the interior of B in A is empty and A is a compact manifold. We illustrate this with a few interesting counterexamples. Finally, we move our focus to locally Euclidean spaces and prove the following result. Theorem 2 Let n ∈ N\{;0}; and let K be a compact subset of Rn such that Rn \ K has finitely many connected components. Then Rn/K has local computable type.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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