# Open Sets with respect to Different Topologies

• Nov 21st 2012, 12:39 PM
Conn
Open Sets with respect to Different Topologies
Hi, just trying to wrap my head around a few concepts; I've only just started topology so correct me if I'm wrong: is it possible, given a set X with two different topologies, to have a subset of X open with regard to one topology, and not open with regard to the other? And could someone give me an example of this please?

Thanks!
• Nov 21st 2012, 12:47 PM
Deveno
Re: Open Sets with respect to Different Topologies
sure.

here is an extreme example:

let τ = 2X, the power set of X. in this topology EVERY subset of X is open (this is called the discrete topology on X).

let σ = {Ø,X}. here there are only TWO open sets, the empty set, and X itself (this is called the indiscrete topology on X).

if X contains more than one element, then the set {x} for a single element of X, is open in the first, but not in the second.
• Nov 21st 2012, 01:43 PM
Plato
Re: Open Sets with respect to Different Topologies
Quote:

Originally Posted by Conn
is it possible, given a set X with two different topologies, to have a subset of X open with regard to one topology, and not open with regard to the other? And could someone give me an example of this please?

Think about. What does it mean for the two typologies to be different?
Keep in mind, the sets of the topology are the open sets.