What is the TOPOLOGICAL DIMENSION of?

[Take_it_Easy]

Let {\mathbb I} = {\mathbb R} \setminus {\mathbb Q} the set of the irrational numbers of the real line.

What is the topological dimension of
{\mathbb R}^2 \setminus {\mathbb I} \times {\mathbb I} ?

 

[Take_it_Easy]

Let {\mathbb I} = {\mathbb R} \setminus {\mathbb Q} the set of the irrational numbers of the real line.

What is the topological dimension of
{\mathbb R}^2 \setminus {\mathbb I} \times {\mathbb I} ?

Just discovered, by myself, that it is 1 .



I just hope that someone can enjoy this result as I do!

 

[net_nubie]

Could you explain what is topological dimension please?

 

[ice109]

apparently

We say a topological space X has topological dimension m if every covering C of X has a refinementC' in which every point of X occurs in at most m+1 sets in C' , and m is the smallest such integer. Actually, this version of the definition of dimension (called the covering dimension) makes the most sense for compact spaces X.