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 aquotient mapprovided 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:

if

if

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