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Tag Archives: continuity
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
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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 εδ 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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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: Limits and Continuity (3)
Let’s consider multivariate functions where . To that end, we need the Euclidean distance function on Rn. If x = (x1, x2, …, xn) is in Rn, we define: Note that x = 0 if and only if x is the zero vector 0. Now we are ready … Continue reading
Posted in Notes
Tagged advanced, analysis, continuity, limits, multivariate, open balls, open subsets, topology
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