Are the following Sets: Open, Closed, Compact, Connected

[emergentecon]

Homework Statement

Ok I created this question to check my thinking.
Are the following Sets: Open, Closed, Compact, Connected

Note: Apologies for bad notation.

S: [0,1)∪(1,2]
V: [0,1)∩(1,2]

Homework Equations

S: [0,1)∪(1,2]
V: [0,1)∩(1,2]

The Attempt at a Solution

S: [0,1)∪(1,2]
Closed - because 0 and 2 represent boundary points
Compact - because S is closed and bounded
Not Connected - because it can be separated into two open disjoint sets [0,1) and [2]

V: [0,1)∩(1,2]
Open - because the set is (1)
Not Compact - because it is open
Connected - because it cannot be separated into two open disjoint sets

 

[MostlyHarmless]

Another definition of closed is that the set contains its limit points. I can think of one limit point of S that is not in S. Are you familiar with the definition of a ball or some books call it a neighborhood.

As for V, as written, V is empty. The ")" means everything up to, but not including.

 

[emergentecon]

Another definition of closed is that the set contains its limit points. I can think of one limit point of S that is not in S. Are you familiar with the definition of a ball or some books call it a neighborhood.

As for V, as written, V is empty. The ")" means everything up to, but not including.

Yes I am aware of a ball. It is used in my book to explain the concept of an open set.
But even so, I thought it was closed, because for the Union I get [0,2] which is, as I understood it, by default closed, given that it includes the boundary points namely, 0 and 2?
What am I missing? What example do you have?As for V . . . didn't realize the intersection there is the null set . . . really thought it was 1 . . . damn.

Thanks.

 

[MostlyHarmless]

You're getting confused on your interval notation. 1 is not in your set S as written. 1 is not in [0, 1) and 1 is not in (1,0]. But any ball centered at 1 will intersect your set non-trivially, thus 1 is a limit point but it is not in S. So S is not closed.

Also, closed and not open are not the same thing, a set can be both closed and open or neither open nor closed.