- #1
PsychonautQQ
- 784
- 10
Let us look at the topological space R_d x R where R_d is the set of real numbers with the discrete toplogy and R the euclidean topology. This set is not second countable, because R_d has no countable basis.
I am wondering if this space is locally euclidean, and if so, of what dimension? Given a point (x,y) of R_d x R, is there a neighborhood that is homeomorphic to a neighborhood of R^n for some n?
Well every point in R_d has a neighborhood homeomorphic to {0} in R^0, and every point in R has a neighborhood homeomorphic to a neighborhood in, well, R. So I think R_d x R is locally euclidean of dimension 1.
I am wondering if this space is locally euclidean, and if so, of what dimension? Given a point (x,y) of R_d x R, is there a neighborhood that is homeomorphic to a neighborhood of R^n for some n?
Well every point in R_d has a neighborhood homeomorphic to {0} in R^0, and every point in R has a neighborhood homeomorphic to a neighborhood in, well, R. So I think R_d x R is locally euclidean of dimension 1.