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Tag Archives: topology
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
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
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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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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
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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
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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
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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
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Tagged advanced, bases, clopen sets, homeomorphisms, metric spaces, subbases, subspaces, topology
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Topology: Bases and Subbases
Bases Recall that though a subring or ideal of a ring may be rather huge, it often suffices to specify just a few elements which will generate the subring or ideal. Likewise, in a topology, one can specify a few … Continue reading
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Tagged bases, furstenberg's proof, generated topology, infinitude of primes, open balls, open subsets, subbases, topology
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Topology: Basic Definitions
Motivation and Definition While studying analysis, one notices that many important concepts can be defined in terms of “open sets”. One gets the inkling that this concept is critical in forming our notions of continuity, limits etc. In this article, we … Continue reading
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Tagged analysis, closed subsets, discrete topology, metric spaces, open balls, open subsets, topology, topoloical equivalence
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Mathematics and Such 