Topology Quotient Space
In Topology 2ed by Munkres, the following definition is given of a quotient map:
Let be topological spaces; let be a surjective map. The map is said to be a quotient map provided a subset is open in if and only if is open in .
Then he gives the following example:
Let be the subspace of . The map defined by:
In order to show that this map is a quotient map, one needs to show that is is surjective, continuous and closed. The first two are clear, however I am not so sure about how to verify that it is a closed map.
Also, he notes that the map is not open, saying as proof that the image of the "open set" of is not open in .
Can someone please tell me:
(i) how to verify that the map is a closed map
(ii) how can be considered open in
(iii) how then is image of not open in
I guess for (ii), I simply say that is , which is clearly closed is open in , but what about (i) and (iii)?
Sorry for the bother, I got it:
(i) is closed in and is also closed in thus is a closed map
(iii) and , which is not closed; thus, is not open in , and so it follows that is not an open map
What is Y? Is it ? Is open in Y?