Difference between revisions of "Topology"
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Let {{M|(X,\mathcal{J})}} and {{M|(X,\mathcal{K})}} be two [[Topological space|topologies]] on {{M|X}} | Let {{M|(X,\mathcal{J})}} and {{M|(X,\mathcal{K})}} be two [[Topological space|topologies]] on {{M|X}} | ||
===Coarser, Smaller, Weaker=== | ===Coarser, Smaller, Weaker=== | ||
| − | Given two topologies <math>\mathcal{J}</math>, <math>\ | + | Given two topologies <math>\mathcal{J}</math>, <math>\mathcal{K}</math> on {{M|X}} we say:<br/> |
<math>\mathcal{J}</math> is '''coarser, smaller''' or '''weaker''' than <math>\mathcal{K}</math> if <math>\mathcal{J}\subset\mathcal{K}</math> | <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. | '''Smaller''' is a good way to remember this as there are 'less things' in the smaller topology. | ||
| − | + | ||
| − | ===Finer | + | ===Finer, Larger, Stronger=== |
Given two topologies <math>\mathcal{J}</math>, <math>\mathcal{K}</math> on {{M|X}} we say:<br/> | Given two topologies <math>\mathcal{J}</math>, <math>\mathcal{K}</math> on {{M|X}} we say:<br/> | ||
<math>\mathcal{J}</math> is '''finer, larger''' or '''stronger''' than <math>\mathcal{K}</math> if <math>\mathcal{J}\supset\mathcal{K}</math> | <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 ' | + | '''Larger''' is a good way to remember this as there are 'more things' in the larger topology. |
==Building new topologies== | ==Building new topologies== | ||
Revision as of 16:25, 23 August 2015
Once you have understood metric spaces you can read motivation for topology and see why topological spaces "make sense" and extend metric spaces.
Contents
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:
- 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
- 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]
- 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 [ilmath]\mathcal{P}(X)[/ilmath] 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]
