Definition of "with dense range"

Question

it is written "T is a continuous map with dense range". What is the definition of map with dense range? Maybe it's simple but I am not native English speaker, thanks.

I think it means that the range is a dense subset. For example, if you have a vector space $V$, a function $T$ has a dense range if $T\langle V \rangle$ is dense, i.e. it's closure is $V$. — Botond, Jul 15 '20 at 13:25

score 2 · Answer 1 · answered Jul 15 '20 at 13:35

2

If $T$ is a continuous map from a topological space $X$ to a topological space $Y$, then $T$ is said to have a dense range if the set $T(X)$ is a dense subset of $Y$. That is, every non-empty open set in $Y$ has a non-empty intersection with $T(X)$.

answered Jul 15 '20 at 13:35

uniquesolution

18,541

Thank you very much :D – Jul 16 '20 at 10:17

Definition of "with dense range"

1 Answers1