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Tag Archives: open balls
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
Posted in Notes
Tagged advanced, closed balls, closed subsets, closures, open balls, points of accumulation, product topology, 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
Posted in Notes
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
Posted in Notes
Tagged analysis, closed subsets, discrete topology, metric spaces, open balls, open subsets, topology, topoloical equivalence
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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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