Topological Spaces#
+ + +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 is the set of open sets. +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.
+Definition 80 (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
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\(\emptyset, X \in \mathcal{U},\)
+If \({U_i}_{i \in I} \subseteq \mathcal{U},\) then \(\bigcup_{i \in I} U_i \in \mathcal{U},\)
+If \(U_1, \dots, U_n \in \mathcal{U},\) then \(\bigcap_{i=1}^n U_i \in \mathcal{U}.\)
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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.
+Definition 81 (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.
+Definition 82 (Continuous function)
+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 \(X.\)
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