Relations between Points and Subsets

Assume given a metric or topological space X and any subset A ⊂ X. We discuss the possible relationships between a point x ∈ X and A in terms of neighborhoods. First Level We note that every neighborhood N of x contains x (so is non-empty). At this level, we make no distinction between x and other points of N. This provides enough information for most of our work.

There are exactly three possibilities, with no overlap:

Case 1. x has a neighborhood N that is contained in A. We call x an interior point of A. Such points form the interior set Int A, or sometimes Å, of A. Since x ∈ N ⊂ A, we have Int A ⊂ A.

Case 2. x has a neighborhood N that is contained in the complement X − A of A. We call x an exterior point of A. Since x ∈ N, these points never lie in A. They form the exterior set Ext A = Int(X −A) of A.

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

To summarize, every point x ∈ X lies in exactly one of the three sets Int A, Ext A, and Fr A. The ambiguity in Case 3 motivates the next definition. Definition 1 We call A closed (in X) if it contains all of its boundary points. We call A open (in X) if it contains none of its boundary points.

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

Lemma 2 For any subset A ⊂ X:

(a)The interior Int A is open; (b)If V ⊂ A is open in X, then V ⊂ Int A. Thus Fr A = X − (Int A ∪ Ext A) = X − (Int A ∪ Int(X −A)) is closed. The closure Cl A = \overline{A} of A may be defined as Int A ∪ Fr A or as A ∪ Fr A. Since

Cl A = X − Ext A, it is closed, and if F is any closed set that contains A, i. e. F ⊃ A, we must have Cl A ⊂ F. Second Level This is more subtle. At this level, we do distinguish between x and other points of N. The limit points and isolated points of A can now be defined. There are now eight possibilities, summarized in the table:

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

110.413 Introduction to Topology JMB File: toprel2, Revision A, 25 Feb 2003