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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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

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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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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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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

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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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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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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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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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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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