Difference between revisions of "Topology"

Revision as of 16:58, 8 March 2015

Once you have understood metric spaces you can read motivation for topology and see why topological spaces "make sense" and extend metric spaces.

Comparing topologies

Let [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](X,\mathcal{K})[/ilmath] be two topologies on [ilmath]X[/ilmath]

Coarser, Smaller, Weaker

Given two topologies [math]\mathcal{J}[/math], [math]\mathcal{K}[/math] on [ilmath]X[/ilmath] we say:
[math]\mathcal{J}[/math] is coarser, smaller or weaker than [math]\mathcal{K}[/math] if [math]\mathcal{J}\subset\mathcal{K}[/math]

Smaller is a good way to remember this as there are 'less things' in the smaller topology.

Finer, Larger, Stronger

Given two topologies [math]\mathcal{J}[/math], [math]\mathcal{K}[/math] on [ilmath]X[/ilmath] we say:
[math]\mathcal{J}[/math] is finer, larger or stronger than [math]\mathcal{K}[/math] if [math]\mathcal{J}\supset\mathcal{K}[/math]

Larger is a good way to remember this as there are 'more things' in the larger topology.

Building new topologies

There are a few common ways to make new topologies from old:

1. Product Given topological spaces [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] there is a topology on [ilmath]X\times Y[/ilmath] called "the product topology" (the coarsest topology such that the projections are continuous
2. Quotient Given a topological space [ilmath](X,\mathcal{J})[/ilmath] and an equivalence relation [ilmath]\sim[/ilmath] on [ilmath]X[/ilmath], we can define the quotient topology on [ilmath]X[/ilmath] which we often denote by [ilmath]\frac{\mathcal{J} }{\sim} [/ilmath]
3. Subspace Given a topological space [ilmath](X,\mathcal{J})[/ilmath] and any [ilmath]Y\subset X[/ilmath] then the topology on [ilmath]X[/ilmath] can induce the subspace topology on [ilmath]Y[/ilmath]

Common topologies

Discreet topology

Given a set [ilmath]X[/ilmath] the Discreet topology on [ilmath]X[/ilmath] is [ilmath]\mathcal{P}(X)[/ilmath], that is [ilmath](X,\mathcal{P}(X))[/ilmath] is the discreet topology on [ilmath]X[/ilmath] where {{\mathcal{P}(X)}} is the power set of [ilmath]X[/ilmath].

That is every subset of [ilmath]X[/ilmath] is an open set of the topology

Indiscreet Topology

Given a set [ilmath]X[/ilmath] the indiscreet topology on [ilmath]X[/ilmath] is the topology [ilmath](X,\{\emptyset,X\})[/ilmath]