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Tag Archives: advanced
Topology: Finite Intersection Property (Omake)
The whole point of this article is the following seemingly trivial observation. Theorem. A topological space X is compact if and only if it satisfies the finite intersection property (F.I.P.): if is a collection of closed subsets of X such that … Continue reading
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Tagged advanced, compact spaces, finite intersection property, four colour theorem, topology
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Topology: More on Compact Spaces
In the previous article, we defined compact spaces as those where every open cover has a finite subcover, i.e. if then we can find a finite set of indices such that On an intuitive level, one should imagine a compact … Continue reading
Topology: Complete Metric Spaces
[ This article was updated on 8 Mar 13; the universal property is now in terms of Cauchy-continuous maps. ] On an intuitive level, a complete metric space is one where there are “no gaps”. Formally, we have: Definition. A … Continue reading
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Tagged advanced, cauchy sequences, complete metric spaces, completion, metric spaces, topology, universal properties
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Topology: Hausdorff Spaces and Dense Subsets
Hausdorff Spaces Recall that we’d like a condition on a topological space X such that if a sequence converges, its limit is unique. A sufficient condition is given by the following: Definition. A topological space X is said to be Hausdorff if … Continue reading
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Tagged advanced, continuity, dense subsets, Hausdorff, metric spaces, nets, topology
4 Comments
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
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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
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Tagged advanced, closed balls, closed subsets, closures, open balls, points of accumulation, product topology, topology
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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
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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
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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
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Tagged advanced, continuity, convergence, extended reals, Hausdorff, limits, sequences, topology
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Mathematics and Such 