Difference between revisions of "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.

Let $(X,\mathcal{J})$ and $(X,\mathcal{K})$ be two topologies on $X$

Given two topologies , on $X$ we say:
$\mathcal{J}$ is coaser, smaller or weaker than $\mathcal{K}$ if $\mathcal{J}\subset\mathcal{K}$

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

Given two topologies , on $X$ we say:
$\mathcal{J}$ is finer, larger or stronger than $\mathcal{K}$ if $\mathcal{J}\supset\mathcal{K}$

Larger is a good way to remember this as there are 'more things' in the larger 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.
+==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>
+'''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 {{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>
+'''Larger''' is a good way to remember this as there are 'more things' in the larger topology.
 [[Category:Topology]]