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Tag Archives: metric spaces
Topology: Sequentially Compact Spaces and Compact Spaces
We’ve arrived at possibly the most confusing notion in topology/analysis. First, we wish to fulfil an earlier promise: to prove that if C is a closed and bounded subset of R and f : R → R is continuous, then f(C) is closed and bounded. [ As … Continue reading
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Tagged compact spaces, metric spaces, nets, sequences, sequentially compact spaces, subnets
4 Comments
Topology: Complete Metric Spaces
[ This article was updated on 8 Mar 13; the universal property is now in terms of Cauchycontinuous maps. ] On an intuitive level, a complete metric space is one where there are “no gaps”. Formally, we have: Definition. A … Continue reading
Posted in Notes
Tagged advanced, cauchy sequences, complete metric spaces, completion, metric spaces, topology, universal properties
10 Comments
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
Posted in Notes
Tagged advanced, continuity, dense subsets, Hausdorff, metric spaces, nets, topology
2 Comments
Topology: Cauchy Sequences and Uniform Continuity
[ Updated on 8 Mar 13 to include Cauchycontinuity 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, cauchycontinuity, 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: 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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