topological properties and nets
For a detailed proof, click here (http://planetmath.org/NetsAndClosuresOfSubspaces).
3 Limit point
Let be a topological space and a subset. A point is a limit point of if and only if there is a net in converging to that is not eventually constant.
A topological space is Hausdorff if and only if every convergent net in has a unique limit.
A topological space is compact if and only if every net in has a convergent subnet.
For a detailed proof, click here (http://planetmath.org/CompactnessAndConvergentSubnets).
For a detailed proof, click here (http://planetmath.org/ContinuityAndConvergentNets).
7 Open map
Let be a surjective map between the topological spaces and . Then is an open mapping if and only if given a net such that , then for every there exists a subnet that to a net such that . By ”” we that is such that .
For a detailed proof, click here (http://planetmath.org/ContinuousSurjectiveOpenMapsInTermsOfNets).
8 Initial topology
Let be a set, a family of topological spaces and a family of functions.
8.0.1 Particular case: subspace topology
8.0.2 Particular case: product topology
9 Compact-open topology
Let be a locally compact Hausdorff space, a topological space and the set of continuous functions from to . A net in converges to in the compact-open topology if and only if whenever a net in , indexed by the same directed set , converges to , we also have that converges to .
|Title||topological properties and nets|
|Date of creation||2013-03-22 18:38:04|
|Last modified on||2013-03-22 18:38:04|
|Last modified by||asteroid (17536)|