Open And Closed Discs - What's The Difference?

[BOAS]

Hello,

i'm working through Lang's 'Introduction To Linear Algebra' and am on page 18 (in case any of you are familiar with it).

He says that the set of points $X$, such that $||X - P|| < a$ where P is a point in the plane and a is a number > 0 is an open disc.

He then goes on to say that $||X - P|| \leq a$ will be the closed disc.

I'm having trouble understanding what the significance of this distinction is. I understand that if the set of points is equal to a, you get a circle, i.e the set of points at a distance a from p (in 2-space at least).

But it seems to me that an open disk is essentially a tiny tiny bit smaller than the close one... So, is there a clear difference that I'm not seeing?

EDIT - Okay, I see the difference. A closed circle is the set of points inside the circle and the circle itself, whilst the open ball is the set of points inside the circle, but not the circle itself.

Feel free to delete this :)

 

[jedishrfu]

Look at the border and you'll see closed means it includes the border points. Okay I see your edit...