Tag Archives: open balls

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: Bases and Subbases

Posted on January 21, 2013 by limsup

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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Topology: Basic Definitions

Posted on January 18, 2013 by limsup

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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Basic Analysis: Limits and Continuity (3)

Posted on November 25, 2012 by limsup

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 →

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