# Continuous map

dapias09
Hi all,

I need help with a paragraph of my book that I don't understand. It says: "the map sending all of ℝ^n into a single point of ℝ^m is an example showing that a continuous map need not send open sets into open sets".

My confusion arising because I can't figure out how this map can be continuous, since the definition is:
" a map is continuous if the inverse image of an open set of the range is an open set". In this case it seems that a single point of R^m isn't an open set, so how can we talk about continuity?

Diego.

SteveL27
Hi all,

I need help with a paragraph of my book that I don't understand. It says: "the map sending all of ℝ^n into a single point of ℝ^m is an example showing that a continuous map need not send open sets into open sets".

My confusion arising because I can't figure out how this map can be continuous, since the definition is:
" a map is continuous if the inverse image of an open set of the range is an open set". In this case it seems that a single point of R^m isn't an open set, so how can we talk about continuity?

Diego.

Let f be such a map. Say f(x) = p. If Y is an open set in ℝ^m, what is its inverse image under Y? Consider two cases:

a) p is an element of Y. Then the inverse image of X is all of ℝ^n, which is open.

b) p is not an element of Y. Then the inverse image of Y is the empty set, which is open.

Either way, the inverse image of an open set is open.