
Recent Posts
Archives
 May 2020
 April 2020
 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: nets
Topology: Sequentially Compact Spaces and Compact Spaces
We’ve arrived at possibly the most confusing notion in topology/analysis. First, we wish to fulfil an earlier promise: to prove that if C is a closed and bounded subset of R and f : R → R is continuous, then f(C) is closed and bounded. [ As … Continue reading
Posted in Notes
Tagged compact spaces, metric spaces, nets, sequences, sequentially compact spaces, subnets
4 Comments
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
Posted in Notes
Tagged advanced, continuity, dense subsets, Hausdorff, metric spaces, nets, topology
4 Comments
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