Talk:Open set
|
|
give example of a set which is both open and closed
-- in euclidean spaces both the empty set and the whole space itself are simultaneously open and closed.
Why is e used instead of ε?
Typo?
Should the "rational" and "real" be switched in the second paragraph?
As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1.
Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers.
- I think the second paragraph is good as it stands. When nothing is mentioned about the surrounding space it is assumed to be the usual topology on the real line.
- Does this answer your question? Oleg Alexandrov 19:23, 18 Mar 2005 (UTC)
Intuitivity
I've always pictured open and closed sets as drawn on paper with a pencil. The boundary is drawn by pressing the pencil in a normal writing angle at the paper, making a clearly visible, narrow, black line. The interior, however, is drawn by holding the pencil almost parallel to the paper and shading the area, making it a fuzzy gray. Closed sets have boundaries around the interior, open sets don't.
This has the added bonus of being able to visualise a separation of a space into an open set and a closed one. The line in the middle has to belong to exactly one set. — JIP | Talk 19:38, 18 Mar 2005 (UTC)