{"id":627,"date":"2018-11-01T12:32:30","date_gmt":"2018-11-01T12:32:30","guid":{"rendered":"https:\/\/konrad.burnik.org\/wordpress\/?p=627"},"modified":"2021-04-24T12:26:21","modified_gmt":"2021-04-24T12:26:21","slug":"metric-bases-for-the-unit-square","status":"publish","type":"post","link":"https:\/\/konrad.burnik.org\/wordpress\/metric-bases-for-the-unit-square\/","title":{"rendered":"Two questions related to computability structures on the unit square"},"content":{"rendered":"

The following questions were proposed to me by professor Zvonko Iljazovi\u0107.<\/p>\n

Let \"I = [0,1]\". We define the metric \"d_\infty\" be the metric defined as<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

for all \"(x_1,x_2), (y_1,y_2) \in I^2\". Let \"\mathcal{M} = (I^2, d_\infty)\".<\/p>\n

Let \"a = (0,0)\" and \"b = (0,1)\". Suppose \"(x_i)\" is a sequence in \"\mathcal{M}\" such that the sequences \"d(x_i, a)\" and \"d(x_i,b)\" are computable sequences  \"\mathbb{N} \rightarrow \mathbb{R}\"? Does the sequence \"(x_i)\" need to be computable in the sense that its components are computable sequences \"\mathbb{N} \rightarrow \mathbb{R}\"?<\/p>\n

\"Rendered<\/p>\n

Let \"w = (1, \gamma)\" where \"0 < \gamma < 1\" is an incomputable real. Then \"d_\infty(x_i, a) = d_\infty(x_i, b)=1\" however, \"(x_i)\" is not a computable sequence.<\/p>\n

Now the second question. Let \"a=(0,0)\", \"b=(0,1)\" and \"c=(1,0)\". Let \"(x_i)\" be a sequence such that \"d_\infty(x_i,a)\", \"d_\infty(x_i,b)\" and \"d_\infty(x_i,c)\" are computable sequences \"\mathbb{N} \rightarrow \mathbb{R}\". Does the sequence \"(x_i)\" need to be computable?<\/p>\n

The answer is yes. The proof is available in our upcoming publication \"Dense Computability Structures\".<\/em><\/p>\n

Proposition<\/strong>:
\nThe metric space \"\mathcal{M}\" has at least two maximal computability structures in which points \"a=(0,0)\" and \"b=(0,1)\" are computable.<\/p>\n

Proof<\/strong>:
\nLet \"(q_j)\" be a computable sequence in \"\mathbb{R}^2\" such that \"\mathrm{Im}\ q = \mathbb{Q}^2 \cap \mathcal{M}\". Let \"S_q\" be a computability structure induced by \"q\". Since \"S_q\" is separable, it is also maximal.<\/p>\n

Let \"c = (1,\gamma)\" where \"\gamma \in \langle 0,1 \rangle\" is an incomputable real. Then the finite sequence \"(a,b,c)\" has the property \"d_\infty(a,b) = d_\infty(a,c) = d_\infty(b,c) = 1\" hence by definition \"(a,b,c)\" is an effective finite sequence.
\nTherefore, <\/p>\n

  <\/span>   <\/span>\"\[T = \{(a,a,a,\dots), (b,b,b,\dots), (c,c,c,\dots)\}\]\"<\/p>\n

is a computability structure and there exists a maximal computability structure \"M\" such that \"T \subseteq M\" and \"a,b,c\" are computable points in \"M\".<\/p>\n

On the other hand, the point \"c\" is not computable in \"S_q\" since that would contradict the fact that \"w\" is a non-computable real. This is equivalent to the fact that \"(c,c,c,\dots) \not \in S_q\". Therefore, \"M \not = S_q\". From this we conclude that \"M\" and \"S_q\" are two maximal computability structures on \"\mathcal{M}\" in which points \"a\" and \"b\" are computable.<\/p>\n

0<\/span><\/a><\/div> <\/div>
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The following questions were proposed to me by professor Zvonko Iljazovi\u0107. Let . We define the metric be the metric defined as     for all . Let . Let and . Suppose is a sequence in such that the sequences and are computable sequences  ? Does the sequence need to be computable in the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"_s2mail":"yes","footnotes":""},"categories":[5],"tags":[],"class_list":["post-627","post","type-post","status-publish","format-standard","hentry","category-computable-metric-spaces"],"_links":{"self":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/627","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/comments?post=627"}],"version-history":[{"count":29,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/627\/revisions"}],"predecessor-version":[{"id":1111,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/627\/revisions\/1111"}],"wp:attachment":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/media?parent=627"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/categories?post=627"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/tags?post=627"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}