# Open sets

• Oct 11th 2009, 01:13 PM
bigdoggy
Open sets
Prove that (0,1] is open in [-1,1] but not in R?
• Oct 11th 2009, 01:22 PM
Matt Westwood
Quote:

Originally Posted by bigdoggy
Prove that (0,1] is open in [-1,1] but not in R?

The point 1 does not have a neighbourhood in R that does not include elements that are not in (0,1].

Too many negatives. Try again.

All neighbourhoods of 1 in R contain elements that are not in (0,1]. However, in [-1,1] this is not the case as there are no elements of [-1,1] that are not in a neighbourhood of 1.

At the other end of the interval (0,1], the end is open. Every point near 0 has a neighbourhood completely contained in (0,1].

So every point of (0,1] has a neighbourhood entirely within (0,1] except 1, for which see above.
• Oct 11th 2009, 03:24 PM
tonio
Quote:

Originally Posted by bigdoggy
Prove that (0,1] is open in [-1,1] but not in R?

Lemma: if X is a topological space and Y is a subset of Y with the inherited topology from X, then a subset A of Y is opne in Y iff A = Y /\ U, with U open in X.

This solves the problem.

Tonio
• Oct 11th 2009, 09:30 PM
Matt Westwood
Quote:

Originally Posted by tonio
Lemma: if X is a topological space and Y is a subset of Y with the inherited topology from X, then a subset A of Y is opne in Y iff A = Y /\ U, with U open in X.

This solves the problem.

Tonio

What's Y /\ U?
• Oct 12th 2009, 01:16 AM
harbottle
I guess that's the intersection.. he means of course that if you use the subspace topology then a set is open if it is the intersection of he subset and an open set in the base set.

eg (0,1] is open in [1,1] because (0,2) is open in R, and (0,2) intersected with [0,1] = (0,1]..
• Oct 12th 2009, 02:20 AM
tonio
Quote:

Originally Posted by Matt Westwood
What's Y /\ U?

Intersection....

Tonio