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_sources/book/topology/002-topological-spaces.md

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+# Topological Spaces
+
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+
+In the previous chapter we saw that in a metric space, continuity of functions is only indirectly determined by the metric itself.
+Instead, the structure that determines continuity {prf:ref}`is the set of open sets<topology:theorem-characterisation-of-continuity>`.
+This motivates the definition of a topological space, which abstracts the notion of open sets from metric spaces.
+
+## Topologies
+First, we define topological spaces.
+These are sets equipped with a topology, a collection of subsets which we _define_ to be open.
+Unlike in metric spaces, where we first defined open balls and then used them to define open sets, here we define open sets directly, and require they satisfy certain properties.
+
+:::{prf:definition} Topological space
+:label: topology:def-topological-space
+A topological space is a set $X,$ called the space, together with a collection $\mathcal{U} \subseteq \mathcal{P}(X)$ of subsets of $X,$ called the topology on $X,$ such that
+
+1. $\emptyset, X \in \mathcal{U},$
+2. If ${U_i}_{i \in I} \subseteq \mathcal{U},$ then $\bigcup_{i \in I} U_i \in \mathcal{U},$
+3. If $U_1, \dots, U_n \in \mathcal{U},$ then $\bigcap_{i=1}^n U_i \in \mathcal{U}.$
+
+The elements of $X$ are called points, and the elements of $\mathcal{U}$ are called open sets.
+:::
+
+
+When working with specific spaces, they will often be already be equipped with a metric.
+We refer to the topology associated with a given metric as the induced topology.
+
+:::{prf:definition} Induced topology
+:label: topology:def-induced-topology
+Let $(X, d)$ be a metric space.
+Then, the topology induced by $d$ is the set of all open sets in $X$ with respect to the metric $d.$
+:::
+
+
+We now also re-define continuity in terms of open sets.
+
+:::{prf:definition} Continuous function
+:label: topology:def-continuous-function-topology
+Let $f: X \to Y$ be a function between topological spaces.
+Then, $f$ is continuous if for every open set $U \subseteq Y,$ the pre-image $f^{-1}(U)$ is an open set in $X.$
+:::
+

book/mira/000-exercises.html

@@ -215,6 +215,7 @@
 </li>
 <li class="toctree-l1 has-children"><a class="reference internal" href="../topology/000-intro.html">Topology</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
 <li class="toctree-l2"><a class="reference internal" href="../topology/001-metric-spaces.html">Metric spaces</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../topology/002-topological-spaces.html">Topological Spaces</a></li>
 </ul>
 </li>
 </ul>

book/mira/000-intro.html

@@ -215,6 +215,7 @@
 </li>
 <li class="toctree-l1 has-children"><a class="reference internal" href="../topology/000-intro.html">Topology</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
 <li class="toctree-l2"><a class="reference internal" href="../topology/001-metric-spaces.html">Metric spaces</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../topology/002-topological-spaces.html">Topological Spaces</a></li>
 </ul>
 </li>
 </ul>

book/mira/001-riemann.html

@@ -215,6 +215,7 @@
 </li>
 <li class="toctree-l1 has-children"><a class="reference internal" href="../topology/000-intro.html">Topology</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
 <li class="toctree-l2"><a class="reference internal" href="../topology/001-metric-spaces.html">Metric spaces</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../topology/002-topological-spaces.html">Topological Spaces</a></li>
 </ul>
 </li>
 </ul>

book/mira/002-measures.html

@@ -215,6 +215,7 @@
 </li>
 <li class="toctree-l1 has-children"><a class="reference internal" href="../topology/000-intro.html">Topology</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
 <li class="toctree-l2"><a class="reference internal" href="../topology/001-metric-spaces.html">Metric spaces</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../topology/002-topological-spaces.html">Topological Spaces</a></li>
 </ul>
 </li>
 </ul>

book/papers/ais/ais.html

@@ -215,6 +215,7 @@
 </li>
 <li class="toctree-l1 has-children"><a class="reference internal" href="../../topology/000-intro.html">Topology</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
 <li class="toctree-l2"><a class="reference internal" href="../../topology/001-metric-spaces.html">Metric spaces</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../topology/002-topological-spaces.html">Topological Spaces</a></li>
 </ul>
 </li>
 </ul>
@@ -648,7 +649,7 @@ <h2>Importance-weighted MCMC<a class="headerlink" href="#importance-weighted-mcm
 So we could, in principle, use MCMC within an importance-weighted estimator to reduce its variance.
 The following algorithm is based on this intuition.</p>
 <div class="proof definition admonition" id="definition-1">
-<p class="admonition-title"><span class="caption-number">Definition 80 </span> (Importance weighted MCMC algorithm)</p>
+<p class="admonition-title"><span class="caption-number">Definition 83 </span> (Importance weighted MCMC algorithm)</p>
 <section class="definition-content" id="proof-content">
 <p>Given a proposal density <span class="math notranslate nohighlight">\(q\)</span>, a target density <span class="math notranslate nohighlight">\(p\)</span> and a sequence of transition kernels <span class="math notranslate nohighlight">\(T_1(x, x'), \dots, T_K(x, x')\)</span> be a sequence of transition kernels such that <span class="math notranslate nohighlight">\(T_k\)</span> leaves <span class="math notranslate nohighlight">\(p\)</span> invariant.
 Sampling <span class="math notranslate nohighlight">\(x_0 \sim q(x)\)</span> followed by</p>
@@ -736,7 +737,7 @@ <h2>Annealed Importance Sampling<a class="headerlink" href="#id2" title="Link to
 <p>These distributions interpolate between <span class="math notranslate nohighlight">\(q\)</span> and <span class="math notranslate nohighlight">\(p\)</span> as we vary <span class="math notranslate nohighlight">\(\beta\)</span>.
 AIS then proceeds in a similar way to the importance weighted MCMC algorithm we highlighted above, except that it requires that each <span class="math notranslate nohighlight">\(T_k\)</span> leaves <span class="math notranslate nohighlight">\(\pi_k\)</span>, instead of <span class="math notranslate nohighlight">\(p\)</span>, invariant.</p>
 <div class="proof definition admonition" id="definition-2">
-<p class="admonition-title"><span class="caption-number">Definition 81 </span> (Annealed Importance Sampling)</p>
+<p class="admonition-title"><span class="caption-number">Definition 84 </span> (Annealed Importance Sampling)</p>
 <section class="definition-content" id="proof-content">
 <p>Given a target density <span class="math notranslate nohighlight">\(p\)</span>, a proposal density <span class="math notranslate nohighlight">\(q\)</span> and a sequence <span class="math notranslate nohighlight">\(0 = \beta_0 \leq \dots \leq \beta_K = 1\)</span>, define</p>
 <div class="math notranslate nohighlight">

book/papers/intro.html

@@ -79,7 +79,7 @@
     <link rel="index" title="Index" href="../../genindex.html" />
     <link rel="search" title="Search" href="../../search.html" />
     <link rel="next" title="Shifted window transformers" href="swin/swin.html" />
-    <link rel="prev" title="Metric spaces" href="../topology/001-metric-spaces.html" />
+    <link rel="prev" title="Topological Spaces" href="../topology/002-topological-spaces.html" />
   <meta name="viewport" content="width=device-width, initial-scale=1"/>
   <meta name="docsearch:language" content="en"/>
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@@ -215,6 +215,7 @@
 </li>
 <li class="toctree-l1 has-children"><a class="reference internal" href="../topology/000-intro.html">Topology</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
 <li class="toctree-l2"><a class="reference internal" href="../topology/001-metric-spaces.html">Metric spaces</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../topology/002-topological-spaces.html">Topological Spaces</a></li>
 </ul>
 </li>
 </ul>
@@ -473,12 +474,12 @@ <h1>Stream of papers<a class="headerlink" href="#stream-of-papers" title="Link t
 
 <div class="prev-next-area">
     <a class="left-prev"
-       href="../topology/001-metric-spaces.html"
+       href="../topology/002-topological-spaces.html"
        title="previous page">
       <i class="fa-solid fa-angle-left"></i>
       <div class="prev-next-info">
         <p class="prev-next-subtitle">previous</p>
-        <p class="prev-next-title">Metric spaces</p>
+        <p class="prev-next-title">Topological Spaces</p>
       </div>
     </a>
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book/papers/num-sde/num-sde.html

@@ -214,6 +214,7 @@
 </li>
 <li class="toctree-l1 has-children"><a class="reference internal" href="../../topology/000-intro.html">Topology</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
 <li class="toctree-l2"><a class="reference internal" href="../../topology/001-metric-spaces.html">Metric spaces</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../topology/002-topological-spaces.html">Topological Spaces</a></li>
 </ul>
 </li>
 </ul>
@@ -466,7 +467,7 @@ <h2>Why stochastic differential equations<a class="headerlink" href="#why-stocha
 <h2>The Wiener process<a class="headerlink" href="#the-wiener-process" title="Link to this heading">#</a></h2>
 <p>In order to define the stochastic component of the transition rule of a stochastic system, we must define an appropriate noise model. The Wiener process is a stochastic process that is commonly used for this purpose.</p>
 <div class="proof definition admonition" id="definition-0">
-<p class="admonition-title"><span class="caption-number">Definition 85 </span> (Wiener process)</p>
+<p class="admonition-title"><span class="caption-number">Definition 88 </span> (Wiener process)</p>
 <section class="definition-content" id="proof-content">
 <p>A standard Wiener process over [0, T] is a random variable <span class="math notranslate nohighlight">\(W(t)\)</span> that depends continuously on <span class="math notranslate nohighlight">\(t \in [0, T]\)</span> and satisfies:</p>
 <ol class="arabic simple">
@@ -649,7 +650,7 @@ <h2>Evaluating a stochastic integral<a class="headerlink" href="#evaluating-a-st
 <h2>Euler-Maruyama method<a class="headerlink" href="#euler-maruyama-method" title="Link to this heading">#</a></h2>
 <p>The Euler-Maruyama method is the analoge of the Euler method for deterministic integrals, applied to the stochastic case.</p>
 <div class="proof definition admonition" id="definition-1">
-<p class="admonition-title"><span class="caption-number">Definition 86 </span> (Euler-Maruyama method)</p>
+<p class="admonition-title"><span class="caption-number">Definition 89 </span> (Euler-Maruyama method)</p>
 <section class="definition-content" id="proof-content">
 <p>Given a scalar SDE with drift and diffusion functions <span class="math notranslate nohighlight">\(f\)</span> and <span class="math notranslate nohighlight">\(g\)</span></p>
 <div class="math notranslate nohighlight">
@@ -769,7 +770,7 @@ <h2>Euler-Maruyama method<a class="headerlink" href="#euler-maruyama-method" tit
 <h2>Strong and weak convergence<a class="headerlink" href="#strong-and-weak-convergence" title="Link to this heading">#</a></h2>
 <p>Since the choice of the number of bins <span class="math notranslate nohighlight">\(N\)</span> of the discretisation affects the accuracy of our method, we are interested in how quickly the approximation converges to the exact solution as a function of <span class="math notranslate nohighlight">\(N\)</span>. To do so, we must first define <em>what convergence means</em> in the stochastic case, which leads us to two disctinct notions of convergence, the strong sence and the weak sense.</p>
 <div class="proof definition admonition" id="definition-2">
-<p class="admonition-title"><span class="caption-number">Definition 87 </span> (Strong convergence)</p>
+<p class="admonition-title"><span class="caption-number">Definition 90 </span> (Strong convergence)</p>
 <section class="definition-content" id="proof-content">
 <p>A method for approximating a stochastic process <span class="math notranslate nohighlight">\(X(t)\)</span> is said to have strong order of convergence <span class="math notranslate nohighlight">\(\gamma\)</span> if there exists a constant such that</p>
 <div class="math notranslate nohighlight">
@@ -779,7 +780,7 @@ <h2>Strong and weak convergence<a class="headerlink" href="#strong-and-weak-conv
 </section>
 </div><p>Strong convergence refers to the rate of convergence of the approximation <span class="math notranslate nohighlight">\(X_n\)</span> to the exact solution <span class="math notranslate nohighlight">\(X(\tau_n)\)</span> as <span class="math notranslate nohighlight">\(\Delta t \to 0\)</span>, in expectation. A weaker condition for convergence is rate at which the expected value of the approximation converges to the true expected value, as <span class="math notranslate nohighlight">\(\Delta t \to 0\)</span>, as given below.</p>
 <div class="proof definition admonition" id="definition-3">
-<p class="admonition-title"><span class="caption-number">Definition 88 </span> (Weak convergence)</p>
+<p class="admonition-title"><span class="caption-number">Definition 91 </span> (Weak convergence)</p>
 <section class="definition-content" id="proof-content">
 <p>A method for approximating a stochastic process <span class="math notranslate nohighlight">\(X(t)\)</span> is said to have weak order of convergence <span class="math notranslate nohighlight">\(\gamma\)</span> if there exists a constant such that</p>
 <div class="math notranslate nohighlight">
@@ -798,7 +799,7 @@ <h2>Strong and weak convergence<a class="headerlink" href="#strong-and-weak-conv
 <h2>Milstein’s higher order method<a class="headerlink" href="#milstein-s-higher-order-method" title="Link to this heading">#</a></h2>
 <p>Just as higher order methods for ODEs exist for obtaining refined estimates of the solution, so do methods for SDEs, such as Milstein’s higher order method.</p>
 <div class="proof definition admonition" id="definition-4">
-<p class="admonition-title"><span class="caption-number">Definition 89 </span> (Milstein’s method)</p>
+<p class="admonition-title"><span class="caption-number">Definition 92 </span> (Milstein’s method)</p>
 <section class="definition-content" id="proof-content">
 <p>Given a scalar SDE with drift and diffusion functions <span class="math notranslate nohighlight">\(f\)</span> and <span class="math notranslate nohighlight">\(g\)</span></p>
 <div class="math notranslate nohighlight">

book/papers/rff/rff.html

@@ -215,6 +215,7 @@
 </li>
 <li class="toctree-l1 has-children"><a class="reference internal" href="../../topology/000-intro.html">Topology</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
 <li class="toctree-l2"><a class="reference internal" href="../../topology/001-metric-spaces.html">Metric spaces</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../topology/002-topological-spaces.html">Topological Spaces</a></li>
 </ul>
 </li>
 </ul>
@@ -513,7 +514,7 @@ <h2>The RFF approximation<a class="headerlink" href="#the-rff-approximation" tit
 <p>This is also an unbiased estimate of the kernel, however its variance is lower than in the <span class="math notranslate nohighlight">\(M = 1\)</span> case, since the variance of the average of the sum of <span class="math notranslate nohighlight">\(K\)</span> i.i.d. random variables is lower than the variance of a single one of the variables.
 We therefore arrive at the following algorithm for estimating <span class="math notranslate nohighlight">\(k\)</span>.</p>
 <div class="proof definition admonition" id="definition-1">
-<p class="admonition-title"><span class="caption-number">Definition 82 </span> (Random Fourier Features)</p>
+<p class="admonition-title"><span class="caption-number">Definition 85 </span> (Random Fourier Features)</p>
 <section class="definition-content" id="proof-content">
 <p>Given a translation invariant kernel <span class="math notranslate nohighlight">\(k\)</span> that is the Fourier transform of a probability measure <span class="math notranslate nohighlight">\(p\)</span>, we have the unbiased real-valued estimator</p>
 <div class="math notranslate nohighlight">

book/papers/score-matching/score-matching.html

@@ -215,6 +215,7 @@
 </li>
 <li class="toctree-l1 has-children"><a class="reference internal" href="../../topology/000-intro.html">Topology</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
 <li class="toctree-l2"><a class="reference internal" href="../../topology/001-metric-spaces.html">Metric spaces</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../topology/002-topological-spaces.html">Topological Spaces</a></li>
 </ul>
 </li>
 </ul>
@@ -470,7 +471,7 @@ <h2>The score matching trick<a class="headerlink" href="#the-score-matching-tric
 <p>The second step is to find a way to use the score function <span class="math notranslate nohighlight">\(\psi_\theta(x)\)</span> along with some observed data, to estimate the parameters <span class="math notranslate nohighlight">\(\theta\)</span>.
 We can achieve this by defining the following score matching objective.</p>
 <div class="proof definition admonition" id="definition-0">
-<p class="admonition-title"><span class="caption-number">Definition 83 </span> (Score matching objective)</p>
+<p class="admonition-title"><span class="caption-number">Definition 86 </span> (Score matching objective)</p>
 <section class="definition-content" id="proof-content">
 <p>Given a data distribution <span class="math notranslate nohighlight">\(p_d(x)\)</span> and an approximating distribution <span class="math notranslate nohighlight">\(p_\theta(x)\)</span> with parameters <span class="math notranslate nohighlight">\(\theta\)</span>, we define the score matching objective as</p>
 <div class="math notranslate nohighlight">