Tag Archives: points of accumulation

Topology: Product Spaces (II)

Posted on February 16, 2013 by limsup

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: Closure

Posted on February 10, 2013 by limsup

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: Nets and Points of Accumulation

Posted on February 5, 2013 by limsup

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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Basic Analysis: Closed Subsets and Uniform Continuity

Posted on January 10, 2013 by limsup

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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Basic Analysis: Limits and Continuity (2)

Posted on November 23, 2012 by limsup

Previously, we defined continuous limits and proved some basic properties. Here, we’ll try to port over more results from the case of limits of sequences. Monotone Convergence Theorem. If f(x) is increasing on the open interval (c, a) and has … Continue reading →

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