Relations between Points and Sets

Assume given a fixed metric space or topological space X and any subset E ⊂ X. We list the possible relationships between a typical point p ∈ X and E in terms of neighborhoods. Every neighborhood N of p contains p (so is non-empty).

First Level At this level, we make no distinction between p and other points of N. This provides enough information for most of our work. There are exactly three possibilities, with no overlap:

Case 1. p has a neighborhood N that is contained in E. We call p an interior point of E. Such points form the interior Int E of E. Since p ∈ N, we have Int E ⊂ E.

Case 2. p has a neighborhood N that is contained in the complement E^{c} = X − E of E. We call p an exterior point of E. Since p ∈ N, these points never lie in E. They form the set Int(X − E).

Case 3. Otherwise, every neighborhood N of p contains both a point of E and a point of X − E. We call p a boundary point or frontier point of E. Such points form the boundary or frontier Bd E of E. These points may or may not lie in E. By symmetry, Bd(X − E) = Bd E.

To summarize, every point p ∈ X lies in exactly one of the three sets Int E, Int(X − E), and Bd E. The ambiguity in Case 3 motivates the main definition.

Definition 1 We call E closed (in X) if it contains all of its boundary points. We call E open (in X) if it contains none of its boundary points.

It is obvious from the symmetry that E is open if and only if X − E is closed.

Lemma 2 For any subset E ⊂ X:

(a)The interior Int E is open; (b)If V ⊂ E is open in X, then V ⊂ Int E.

Thus Bd E = X − (Int E ∪ Int(X − E)) is closed.

The closure Cl E = \overline{E} of E may be defined as Int E ∪ Bd E or as E ∪ Bd E. Since \overline{E} = X − Int(X − E), it is closed, and if F is any closed set that contains E, i. e. F ⊃ E, we must have \overline{E} ⊂ F.

Second Level This is more subtle. At this level, we do distinguish between p and other points of N. The limit points and isolated points of E can now be defined. There are now eight possibilities, summarized in the table:

Case Description  in E? int? limit point?
1. p has a neighborhood N with  p ∈ E int isolated
N − p empty.  p ∉ E ext exterior
2. p has a neighborhood N with  p ∈ E int limit
non-empty N − p ⊂ E.  p ∉ E bd limit
3. p has a neighborhood N with  p ∈ E bd isolated
non-empty N − p ⊂ X − E.  p ∉ E ext exterior
4. None of the above: every N − p  p ∈ E bd limit
contains points of E and of X − E. p ∉ E bd limit

110.405 Analysis I JMB File: toprel, Revision B, 1 Mar 2002