{"id":447,"date":"2018-10-28T15:34:52","date_gmt":"2018-10-28T15:34:52","guid":{"rendered":"https:\/\/konrad.burnik.org\/wordpress\/?p=447"},"modified":"2018-11-05T17:39:59","modified_gmt":"2018-11-05T17:39:59","slug":"finite-metric-bases-and-computability-structures","status":"publish","type":"post","link":"https:\/\/konrad.burnik.org\/wordpress\/finite-metric-bases-and-computability-structures\/","title":{"rendered":"Finite metric bases and computability structures"},"content":{"rendered":"

Introduction<\/h2>\n

In July 2018 professor Zvonko Iljazovi\u0107 and myself have attended the CCA 2018 conference<\/a> and presented our joint work titled \"Effective compactness and uniqueness of maximal computability structures\"<\/em>. The original presentation can be found here<\/a>.<\/p>\n

Computability structures<\/h2>\n

In the following I will mention the elements of the theory of computability structures needed to state our main result. These notions are already well known, the main contributing articles for this theory that were used for our work are listed in the references.<\/p>\n

Let \"(X,d)\" be a metric space and \"(x_i)\" a sequence in \"X\". We say \"(x_i)\" is an effective sequence<\/em> in \"(X, d)\" if the function \"\mathbb{N}^2 \rightarrow \mathbb{R}\"<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

is recursive.<\/p>\n

A finite sequence \"x_0,\dots,x_n\" is an effective finite sequence<\/em> if \"d(x_i, x_j)\" is a recursive real number for each \"i,j \in \{0, \dots, n\}\".<\/p>\n

If \"(x_i)\" and \"(y_j)\" are sequences in \"X\", we say \"((x_i), (y_j))\" is an effective pair<\/em> in \"(X,d)\" and write \"(x_i) \diamond (y_j)\" if the function \"\mathbb{N}^2 \rightarrow \mathbb{R}\",<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

is recursive.<\/p>\n

Let \"(X,d)\" be a metric space and \"(x_i)\" a sequence in \"X\". A sequence \"(y_i)\" is computable w.r.t \"(x_i)\"<\/em> in \"(X,d)\" iff there exists a computable \"F:\mathbb{N}^2\rightarrow \mathbb{N}\" such that<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

for all \"i,k \in \mathbb{N}\". We write \"(x_i) \preceq (y_j)\".<\/p>\n

Let \"(X, d)\" be a metric space. A set \"\mathcal{S} \subseteq X^\mathbb{N}\" is a computability structure<\/em> on \"(X,d)\" if the following holds:<\/span><\/p>\n