Determining which sets are open?

Thread starter ~Death~
Start date Sep 26, 2009
Tags

Sets

In summary, a set is open if every point in the set has a neighborhood contained entirely within the set and does not contain its boundary points. A set can be both open and closed, known as a clopen set. The concept of open sets is fundamental in topology and allows for the definition of topological spaces and continuity of functions. An example of a set that is not open is the set of integers, as it does not contain all of its boundary points.

Sep 26, 2009

~Death~

(1/2,1/3),[1/2,1],[0,1/2]

which are open in the subspace topology of the subspace [0,1] of the lower limit topology on R

Mathematics news on Phys.org

Sep 27, 2009

HallsofIvy

Science Advisor

Homework Helper

Use the definition! Which of those is the intersection of [0, 1] with a set open in R (using the lower limit topology, of course)?

1. How do you determine if a set is open?

To determine if a set is open, you must check if every point in the set has a neighborhood contained entirely within the set.

2. What does it mean for a set to be open?

A set is considered open if it does not contain its boundary points. In other words, every point in the set has a neighborhood that is completely contained within the set.

3. Can a set be both open and closed?

Yes, a set can be both open and closed. This is known as a clopen set. An example of a clopen set is the set of all real numbers between 0 and 1, including 0 and 1 themselves.

4. How does the concept of open sets relate to topology?

The concept of open sets is fundamental in topology, as it allows for the definition of topological spaces and continuity of functions. Topology studies the properties of spaces that are preserved under continuous deformations, and open sets play a crucial role in defining these continuous deformations.

5. Can you give an example of a set that is not open?

Yes, the set of integers is an example of a set that is not open. This is because every neighborhood of an integer contains points that are not integers, meaning the set does not contain all of its boundary points.