Let , let be the subspace

Prove is a closed map but not an open map.

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- Jan 15th 2009, 01:06 PMGenoaTopologistsubspace, closed map
Let , let be the subspace

Prove is a closed map but not an open map. - Jan 15th 2009, 04:35 PMMathstud28
- Jan 15th 2009, 04:55 PMchabmgph
Though I am not used to this notation either, I am guessing it is to prove that

the function (which behaviors like but restricted in the subspace ) maps closed sets to closed sets but not open sets to open sets in . (Thinking) - Jan 15th 2009, 09:10 PMMathstud28
- Jan 16th 2009, 12:45 AMMoo
- Jan 16th 2009, 04:25 AMOpalg
The space X is just the union of the two coordinate axes. The mapping f (projection onto the first coordinate) acts as the identity on the x-axis, and takes everything on the y-axis to the origin. You should be able to see from that that the image under f of a closed set will always be closed (think of the set as the union of two parts, the part on the x-axis and the part on the y-axis: the image under f of each of these two parts is fairly obviously closed).

The map is not open, because for example the image of the open subset of X is a single point and therefore not open. - Jan 17th 2009, 02:54 PMMathstud28