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Author Archives: limsup
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: Closure
Suppose Y is a subset of a topological space X. We define cl(Y) to be the “smallest” closed subset containing Y. Its formal definition is as follows. Let Σ be the collection of all closed subsets containing Y. Note that , so Σ is not empty. … Continue reading
Posted in Notes
Tagged advanced, closed balls, closed subsets, closures, open balls, points of accumulation, product topology, topology
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Thoughts on a Problem III
I saw an interesting problem recently and can’t resist writing it up. The thought process for this problem was exceedingly unusual as you’ll see later. First, here’s the source: But here’s the full problem (rephrased a little) if you’d rather … Continue reading
Topology: Cauchy Sequences and Uniform Continuity
[ Updated on 8 Mar 13 to include Cauchy-continuity and added answers to exercises. ] We wish to generalise the concept of Cauchy sequences to metric spaces. Recall that on an intuitive level, a Cauchy sequence is one where the … Continue reading
Posted in Notes
Tagged advanced, cauchy sequences, cauchy-continuity, metric spaces, product topology, topology, uniform continuity
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Topology: Nets and Points of Accumulation
Recall that a sequence in a topological space X converges to a in X if the function f : N* → X which takes is continuous at . Unrolling the definition, it means that for any open subset U of X containing a, the set contains (N, ∞] for some N. In … Continue reading
Posted in Notes
Tagged advanced, closed subsets, continuity, convergence, limits, metric spaces, nets, points of accumulation, sequences, topology
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Topology: Limits and Convergence
Following what we did for real analysis, we have the following definition of limits. Definition of Limits. Let X, Y be topological spaces and . If f : X-{a} → Y is a function, then we write if the function: is … Continue reading
Posted in Notes
Tagged advanced, continuity, convergence, extended reals, Hausdorff, limits, sequences, 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 ε-δ 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: Disjoint Unions
Disjoint Unions Let X and Y be topological spaces and be a set-theoretic disjoint union. We wish to define a topology on Z in a most natural way. Definition. The topology on is defined to be: It’s almost trivial to check that this … Continue reading
Posted in Notes
Tagged advanced, bases, connected spaces, disjoint union topology, metrisable topology, product topology, sub-bases, 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 set-theoretic 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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Mathematics and Such 