Difference between revisions of "Topology"
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Once you have understood [[Metric space|metric spaces]] you can read [[Motivation for topology|motivation for topology]] and see why [[Topological space|topological spaces]] "make sense" and extend metric spaces. | Once you have understood [[Metric space|metric spaces]] you can read [[Motivation for topology|motivation for topology]] and see why [[Topological space|topological spaces]] "make sense" and extend metric spaces. | ||
| − | == | + | ==Comparing topologies== |
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=== |
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 ''' | + | <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. | ||
| Line 14: | Line 14: | ||
'''Larger''' is a good way to remember this as there are 'more things' in the larger topology. | '''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 topology|Product]] Given topological spaces {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} there is a topology on {{M|X\times Y}} called "the product topology" (the coarsest topology such that the [[Projection map|projections]] are continuous | ||
| + | # [[Quotient topology|Quotient]] Given a topological space {{M|(X,\mathcal{J})}} and an [[Equivalence relation|equivalence relation]] {{M|\sim}} on {{M|X}}, we can define the quotient topology on {{M|X}} which we often denote by {{M|\frac{\mathcal{J} }{\sim} }} | ||
| + | # [[Subspace topology|Subspace]] Given a topological space {{M|(X,\mathcal{J})}} and any {{M|Y\subset X}} then the topology on {{M|X}} can induce the subspace topology on {{M|Y}} | ||
| + | |||
| + | ==Common topologies== | ||
| + | ===Discreet topology=== | ||
| + | Given a set {{M|X}} the Discreet topology on {{M|X}} is {{M|\mathcal{P}(X)}}, that is {{M|(X,\mathcal{P}(X))}} is the discreet topology on {{M|X}} where {{\mathcal{P}(X)}} is the [[Power set|power set]] of {{M|X}}. | ||
| + | |||
| + | That is ''every'' subset of {{M|X}} is an open set of the topology | ||
| + | |||
| + | ===Indiscreet Topology=== | ||
| + | Given a set {{M|X}} the indiscreet topology on {{M|X}} is the topology {{M|(X,\{\emptyset,X\})}} | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Revision as of 19:07, 27 February 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 {{\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]
