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Tag Archives: advanced
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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Burnside’s Lemma and Polya Enumeration Theorem (1)
[ Note: this article assumes you know some rudimentary theory of group actions. ] Let’s consider the following combinatorial problem. Problem. ABC is a given equilateral triangle. We wish to colour each of the three vertices A, B and C by … Continue reading
Basic Analysis: Closed Subsets and Uniform Continuity
Let’s consider another question: suppose f : D → R is continuous, where D is a subset of R. If (xn) is a sequence in D converging to some real L, is it true that (f(xn)) is also convergent? Now if L is in D, then we know that (f(xn)) → (f(L)). … Continue reading
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Tagged advanced, analysis, closed subsets, continuity, points of accumulation, uniform continuity, uniform convergence
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Basic Analysis: Uniform Convergence
Once again, let be a subset. Suppose we now have a sequence of functions , where n = 1, 2, 3, … , such that for each x in D, the sequence converges to some real value. We’ll denote this value by f(x), thus … Continue reading
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Tagged advanced, analysis, continuity, convergence, pointwise convergence, series, uniform convergence
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Basic Analysis: Differentiation (2)
Finding Extremum Points One of the most common applications of differentiation is in finding all local maximum and minimum points. Definition. We say f(x) has a local maximum (resp. minimum) at x=a, if there is an open interval (b, c) containing a, such … Continue reading
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Tagged advanced, analysis, compact sets, differentiation, mean value theorem, rolle's theorem
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Basic Analysis: Differentiation (1)
In this article, we’ll look at differentiation more rigourously and carefully. Throughout this article, we suppose f is a real-valued function defined on an open interval (b, c) containing a, i.e. f : (b, c) → R with b < a < c. Theorem. The derivative of f(x) at a is … Continue reading
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Tagged advanced, analysis, analytic, chain rule, differentiation, taylor series
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Mathematics and Such 