Determining Basis for Eclidean Topology on R Squared

[Iuriano Ainati]

In the topic of the topology, how to determine whether or not these collections is the basis for the Eclidean topology, on R squared.

1. the collection of all open squares with sides parallel to the axes.

2. the collection of all open discs.
3. the collection of all open rectangle.
4. the collection of all open triangles.

 

[Hurkyl]

What have you tried? Where are you stuck?

 

[HallsofIvy]

Start by writing out the definition of "basis for a topology".

Then see which of those satisfy the conditions in the definition!

 

[StatusX]

How have you defined the topology for R²? Usually you do it by setting the open disks as a basis (ie, considering it as a metric space with the usual metric), or else considering it as a product space of R (with the open intervals as a basis for R), which would make the open rectangles your basis. Since you're asked to show both of these is a valid basis, I'm curious what else you'd use to do this.