{"id":773,"date":"2018-12-05T17:11:02","date_gmt":"2018-12-05T17:11:02","guid":{"rendered":"https:\/\/konrad.burnik.org\/wordpress\/?p=773"},"modified":"2024-08-04T20:23:19","modified_gmt":"2024-08-04T20:23:19","slug":"investigations-into-infinite-metric-bases-part-2","status":"publish","type":"post","link":"https:\/\/konrad.burnik.org\/wordpress\/investigations-into-infinite-metric-bases-part-2\/","title":{"rendered":"Investigations into infinite metric bases (Part 2\/2)"},"content":{"rendered":"

<\/p>\n

In Part 1<\/a> I posed the following question:<\/p>\n

\"Does there exist a metric space \"(X,d)\" such that it has an infinite metric base \"A\" with the following properties:<\/em><\/p>\n

    \n
  1. \"A\" is not dense;<\/em><\/li>\n
  2. for all finite metric bases \"B\" of \"(X,d)\" we have \"B \not \subseteq A\"?\"<\/em><\/li>\n<\/ol>\n

    Let \"d:\mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}\" be defined as<\/p>\n

      <\/span>   <\/span>\"\[<\/p><\/p>\n

    for all \"(x_1, x_2), (y_1, y_2) \in \mathbb{R}^2\". Then \"d\" is a metric on \"\mathbb{R}^2\", denoted by \"d_\infty\" and \"(\mathbb{R}^2, d_\infty)\" is a metric space.<\/p>\n

    After a brief chat with professor Zvonko Iljazovi\u0107, he proposed the metric space \"(\mathbb{R}^2,d_\infty)\" and the integer lattice<\/a> \"\mathbb{Z}^2\" as the candidate for such a set. Here I will explain why this set is a good candidate.<\/p>\n

    Let \"A = \mathbb{Z}^2\". Then \"A\" is a countable set, which is not dense in \"\mathbb{R}^2\" which can be seen by taking for example \"x = (1/2,1/2)\" and \"\epsilon = 1/4\" then \"B(x,\epsilon) \cap A = \emptyset\". By [1] the space\u00a0\"(\mathbb{R}^2, d_\infty)\" does not have a finite metric base, therefore \"A\" itself can not contain a finite metric base. What is left to prove is that \"\mathbb{Z}^2\" is a metric base for \"(\mathbb{R}^2, d_\infty)\". The rest of this blog post is concerned with proving this fact.<\/p>\n

    First, note that the following result was mentioned briefly in our CCA 2018 presentation<\/a>.<\/p>\n

    Proposition 1<\/strong>. Let \"\{a,b,c\} \in [0,1]\times[0,1]\" such that either \"a=(0,0)\", \"b = (1,1)\" or \"a=(0,1)\", \"b = (1,0).\" Let \"c \not \in \overline{ab}\". Then \"\{a,b,c\}\" is a metric base for \"([0,1]\times[0,1], d_\infty)\".<\/p>\n

    Interestingly enough, the article [1] already gives a complete characterisation of finite metric bases for squares in the digital plane for a couple of metrics including \"d_\infty\". My conjecture is that this also gives a complete characterisation of finite metric bases for \"([0,1]\times[0,1], d_\infty)\" however, this is a topic for another blog post. For now, it seems that the result stated in Proposition 1 coincides with some parts of results from [1] and as such will be sufficient for our needs.<\/p>\n

    We will also need the following claim, which is straightforward to establish.<\/p>\n

    Proposition 2<\/strong>. Let \"(X,d_X)\" and \"(Y,d_Y)\" be metric spaces. If \"\{a_0,\dots,a_n\}\" is a metric base for \"(X, d)\" and \"\phi : X \rightarrow Y\" is an onto mapping such that there exists \"\lambda > 0\" with the property<\/p>\n

      <\/span>   <\/span>\"\[<\/p><\/p>\n

    for all \"x, y \in X\", then \"\{\phi(a_0),\dots,\phi(a_n)\}\" is a metric base for \"(Y, d_Y)\".<\/p>\n

    By using Proposition 2 we can now easily find a metric base for an arbitrary square \"[\alpha,\beta]\times[\alpha,\beta]\" where \"\alpha< \beta\" by using the known result of Proposition 1 and finding a suitable map \"\phi : [0,1]\times[0,1]\rightarrow [\alpha,\beta]\times[\alpha,\beta]\".<\/p>\n

    Proposition 3<\/strong>. Let \"\{a,b,c\}\" be a metric base for \"([0,1]\times[0,1], d_\infty)\". Let \"\phi :[0,1]\times[0,1] \rightarrow [\alpha,\beta]\times[\alpha,\beta]\" where \"\alpha < \beta\" be defined as<\/p>\n

      <\/span>   <\/span>\"\[ \phi(x_1,x_2) = ((1-x_1)\alpha + x_1 \beta, (1-x_2)\alpha + x_2 \beta)) \]\"<\/p><\/p>\n

    for all \"(x_1,x_2) \in [0,1]\times[0,1]\". Then \"\{\phi(a), \phi(b), \phi(c)\}\" is a metric base for \"([\alpha,\beta]\times[\alpha,\beta], d_\infty)\".<\/p>\n

    Note that by setting \"\lambda = \beta - \alpha\" it is easy to verify that \"\phi\" defined in Proposition 3 satisfies the property of \"\phi\" from Proposition 2.<\/p>\n

    Finally, we now have everything we need to prove that \"\mathbb{Z}^2\" is indeed a metric base for \"(\mathbb{R}^2, d_\infty)\".<\/p>\n

    Proposition 4<\/strong>. \"\mathbb{Z}^2\" is a metric base for \"(\mathbb{R}^2, d_\infty)\".<\/p>\n

    Proof<\/em>.<\/p>\n

    Let \"A =\mathbb{Z}^2\". Let \"x, y \in \mathbb{R}^2\" and assume that<\/p>\n

      <\/span>   <\/span>\"\[<\/p><\/p>\n

    for all \"a \in A\".<\/p>\n

    Note that we can always choose \"\alpha<\beta\" such that \"\alpha,\beta\in \mathbb{Z}^2\" and \"x, y \in [\alpha,\beta]\times[\alpha,\beta]\". Namely, we can take for example \"M \in \mathbb{N}\" such that \"M > \max(|x_1|,|x_2|,|y_1|,|y_2|)+1\" and set \"S =\widehat{B}_\infty(0, M)\". Now set \"\alpha = -M\" and \"\beta = M\".<\/p>\n

    From Proposition 1, it follows that any three corners of the square \"[0,1]\times[0,1]\" are a metric base for \"([0,1]\times[0,1], d_\infty)\". By Proposition 3 there is a map \"\phi :[0,1]\times[0,1] \rightarrow [\alpha,\beta]\times[\alpha,\beta]\" such that \"\{\phi(a), \phi(b), \phi(c)\}\" is a metric base for \"[\alpha,\beta]\times[\alpha,\beta]\".\u00a0Since \"\{\phi(a), \phi(b), \phi(c)\} \subset A\", then \"d_\infty(x,p) = d_\infty(y,p)\" for all \"p\in \{\phi(a), \phi(b), \phi(c)\}\" implies \"x = y\".<\/p>\n

    Therefore, \"A\" is a metric base for \"(\mathbb{R}^2, d_\infty)\".<\/p>\n

    Q.E.D.<\/em><\/p>\n

    References<\/h2>\n
      \n
    1. Robert A.\u00a0<\/span>Melter and\u00a0<\/span><\/span>Ioan\u00a0<\/span><\/span>Tomescu<\/span><\/span>\u00a0\"<\/span>Metric Bases in Digital Geometry\",\u00a0Computer Vision, Graphics, and Image Processing Volume 25, Issue 1, January 1984, Pages 113-121 DOI:\u00a0<\/em>https:\/\/doi.org\/10.1016\/0734-189X(84)90051-3<\/a><\/li>\n
    2. Konrad Burnik, Zvonko Iljazovi\u0107, \"Effective compactness and uniqueness of maximal computability structures\", <\/em>CCA 2018 presentation slides:<\/li>\n<\/ol>\n\nEffective_compactness_and_uniqueness_of<\/a>\n\n\n

      Copyright \u00a9 2018, Konrad Burnik<\/p>\n

      0<\/span><\/a><\/div> <\/div>
      <\/div><\/div>
      <\/div>","protected":false},"excerpt":{"rendered":"

      In Part 1 I posed the following question: \"Does there exist a metric space such that it has an infinite metric base with the following properties: is not dense; for all finite metric bases of we have ?\" Let be defined as     for all . Then is a metric on , denoted by […]<\/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":[8],"tags":[],"class_list":["post-773","post","type-post","status-publish","format-standard","hentry","category-metric-spaces"],"_links":{"self":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/773","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=773"}],"version-history":[{"count":89,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/773\/revisions"}],"predecessor-version":[{"id":1299,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/posts\/773\/revisions\/1299"}],"wp:attachment":[{"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/media?parent=773"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/categories?post=773"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/konrad.burnik.org\/wordpress\/wp-json\/wp\/v2\/tags?post=773"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}