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Tag Archives: subspaces
Topology: Product Spaces (II)
The Box Topology Following an earlier article on products of two topological spaces, we’ll now talk about a product of possibly infinitely many topological spaces. Suppose is a collection of topological spaces indexed by I, and we wish to define … Continue reading
Topology: Interior
Let Y be a subset of a topological space X. In the previous article, we defined the closure of Y as the smallest closed subset of X containing Y. Dually, we shall now define the interior of Y to be the largest open subset contained in … Continue reading
Posted in Notes
Tagged advanced, boundaries, closures, interiors, product topology, subspaces, topology
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Topology: Continuous Maps
Continuity in Metric Spaces Following the case of real analysis, let’s define continuous functions via the usual εδ definition. Definition. Let (X, d) and (Y, d’) be two metric spaces. A function f : X → Y is said to be … Continue reading
Posted in Notes
Tagged advanced, continuity, disjoint union topology, homeomorphism, metric spaces, product topology, subspaces, topology
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Topology: Product Spaces (I)
In this article, we consider the product of two topological spaces. To motivate our definition, we first begin with metric spaces (X, dX) and (Y, dY). Letting Z = X × Y be the settheoretic product, we wish to define a metric on Z from dX and … Continue reading
Posted in Notes
Tagged advanced, metric spaces, product topology, subspaces, topology, torus
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Topology: Subspaces
First, suppose (X, d) is a metric space. If Y is a subset of X, then one can restrict the metric to , i.e. for any , we set d’(y, y’) := d(y, y’). It’s not hard to show that the resulting function is a metric on Y. … Continue reading
Posted in Notes
Tagged advanced, bases, clopen sets, homeomorphisms, metric spaces, subbases, subspaces, topology
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