
Recent Posts
Archives
 March 2020
 June 2018
 July 2016
 June 2016
 May 2016
 March 2015
 February 2015
 January 2015
 December 2014
 December 2013
 November 2013
 July 2013
 June 2013
 May 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
 April 2012
 March 2012
 February 2012
 January 2012
 December 2011
 November 2011
 October 2011
Categories
Meta
Pages
Tag Archives: closed subsets
Commutative Algebra 4
More Concepts in Algebraic Geometry Recall that we have a bijection between radical ideals of and closed subsets of . The bijection reverses the inclusion so if and only if . Not too surprisingly, operations on ideals translate to operations … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, closed subsets, ideals, irreducible spaces, rings, zariski topology
Leave a comment
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
Leave a comment
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
Posted in Notes
Tagged advanced, closed subsets, continuity, convergence, limits, metric spaces, nets, points of accumulation, sequences, topology
Leave a comment
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
Leave a comment
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
Posted in Notes
Tagged advanced, analysis, closed subsets, continuity, points of accumulation, uniform continuity, uniform convergence
Leave a comment