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. | ||
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| + | ==Phrases== | ||
| + | Let {{M|(X,\mathcal{J})}} and {{M|(X,\mathcal{K})}} be two [[Topological space|topologies]] on {{M|X}} | ||
| + | ===Coaser, Smaller, Weaker=== | ||
| + | Given two topologies <math>\mathcal{J}</math>, <math>\mathcal{K}</math> on {{M|X}} we say:<br/> | ||
| + | <math>\mathcal{J}</math> is '''coaser, smaller''' or '''weaker''' than <math>\mathcal{K}</math> if <math>\mathcal{J}\subset\mathcal{K}</math> | ||
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| + | '''Smaller''' is a good way to remember this as there are 'less things' in the smaller topology. | ||
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| + | ===Finer, Larger, Stronger=== | ||
| + | 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> | ||
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| + | '''Larger''' is a good way to remember this as there are 'more things' in the larger topology. | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Revision as of 18:45, 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.
Phrases
Let [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](X,\mathcal{K})[/ilmath] be two topologies on [ilmath]X[/ilmath]
Coaser, Smaller, Weaker
Given two topologies [math]\mathcal{J}[/math], [math]\mathcal{K}[/math] on [ilmath]X[/ilmath] we say:
[math]\mathcal{J}[/math] is coaser, 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.
