Tag Archives: metric spaces

Commutative Algebra 54

Posted on May 12, 2020 by limsup

Filtered Rings Definition. Let A be a ring. A filtration on A is a sequence of additive subgroups such that for any . A filtered ring is a ring with a designated filtration. Note Since , in fact each is … Continue reading →

Posted in Advanced Algebra | Tagged completions, filtrations, formal power series, limits, metric spaces, p-adic, ultrametric | 5 Comments

Topology: Sequentially Compact Spaces and Compact Spaces

Posted on February 24, 2013 by limsup

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: Complete Metric Spaces

Posted on February 21, 2013 by limsup

[ This article was updated on 8 Mar 13; the universal property is now in terms of Cauchy-continuous maps. ] On an intuitive level, a complete metric space is one where there are “no gaps”. Formally, we have: Definition. A … Continue reading →

Posted in Notes | Tagged advanced, cauchy sequences, complete metric spaces, completion, metric spaces, topology, universal properties | 11 Comments

Topology: Hausdorff Spaces and Dense Subsets

Posted on February 18, 2013 by limsup

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: Cauchy Sequences and Uniform Continuity

Posted on February 7, 2013 by limsup

[ Updated on 8 Mar 13 to include Cauchy-continuity and added answers to exercises. ] We wish to generalise the concept of Cauchy sequences to metric spaces. Recall that on an intuitive level, a Cauchy sequence is one where the … Continue reading →

Posted in Notes | Tagged advanced, cauchy sequences, cauchy-continuity, metric spaces, product topology, topology, uniform continuity | Leave a comment

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 →

Posted in Notes | Tagged advanced, closed subsets, continuity, convergence, limits, metric spaces, nets, points of accumulation, sequences, topology | Leave a comment

Topology: Continuous Maps

Posted on January 31, 2013 by limsup

Continuity in Metric Spaces Following the case of real analysis, let’s define continuous functions via the usual ε-δ definition. Definition. Let (X, d) and (Y, d’) be two metric spaces. A function f : X → Y is said to be … Continue reading →

Posted in Notes | Tagged advanced, continuity, disjoint union topology, homeomorphism, metric spaces, product topology, subspaces, topology | Leave a comment

Topology: Product Spaces (I)

Posted on January 27, 2013 by limsup

In this article, we consider the product of two topological spaces. To motivate our definition, we first begin with metric spaces (X, dX) and (Y, dY). Letting Z = X × Y be the set-theoretic product, we wish to define a metric on Z from dX and … Continue reading →

Posted in Notes | Tagged advanced, metric spaces, product topology, subspaces, topology, torus | Leave a comment

Topology: Subspaces

Posted on January 25, 2013 by limsup

First, suppose (X, d) is a metric space. If Y is a subset of X, then one can restrict the metric to , i.e. for any , we set d’(y, y’) := d(y, y’). It’s not hard to show that the resulting function is a metric on Y. … Continue reading →

Posted in Notes | Tagged advanced, bases, clopen sets, homeomorphisms, metric spaces, subbases, subspaces, topology | Leave a comment

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 →

Posted in Notes | Tagged analysis, closed subsets, discrete topology, metric spaces, open balls, open subsets, topology, topoloical equivalence | Leave a comment