Interior (topology)

In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.

The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).

If \(S\) is a subset of a Euclidean space, then \(x\) is an interior point of \(S\) if there exists an open ball centered at \(x\) which is completely contained in \(S.\) (This is illustrated in the introductory section to this article.)

This definition generalizes to any subset \(S\) of a metric space \(X\) with metric \(d\): \(x\) is an interior point of \(S\) if there exists a real number \(r > 0,\) such that \(y\) is in \(S\) whenever the distance d(x, y) < r.\)

This definition generalizes to topological spaces by replacing "open ball" with "open set". If \(S\) is a subset of a topological space \(X\) then \(x\) is an interior point of \(S\) in \(X\) if \(x\) is contained in an open subset of \(X\) that is completely contained in \(S.\) (Equivalently, \(x\) is an interior point of \(S\) if \(S\) is a neighbourhood of \(x.\))

The interior of a subset \(S\) of a topological space \(X,\) denoted by \(\operatorname{int}_X S\) or \(\operatorname{int} S\) or \(S^\circ,\) can be defined in any of the following equivalent ways:

1. \(\operatorname{int} S\) is the largest open subset of \(X\) contained in \(S.\)
2. \(\operatorname{int} S\) is the union of all open sets of \(X\) contained in \(S.\)
3. \(\operatorname{int} S\) is the set of all interior points of \(S.\)

If the space \(X\) is understood from context then the shorter notation \(\operatorname{int} S\) is usually preferred to \(\operatorname{int}_X S.\)

• In any space, the interior of the empty set is the empty set.
• In any space \(X,\) if \(S \subseteq X,\) then \(\operatorname{int} S \subseteq S.\)
• If \(X\) is the real line \(\Reals\) (with the standard topology), then \(\operatorname{int} ([0, 1]) = (0, 1)\) whereas the interior of the set \(\Q\) of rational numbers is empty: \(\operatorname{int} \Q = \varnothing.\)
• If \(X\) is the complex plane \(\Complex,\) then \(\operatorname{int} (\{z \in \Complex : |z| \leq 1\}) = \{z \in \Complex : |z| < 1\}.\)
• In any Euclidean space, the interior of any finite set is the empty set.

On the set of real numbers, one can put other topologies rather than the standard one:

• If \(X\) is the real numbers \(\Reals\) with the lower limit topology, then \(\operatorname{int} ([0, 1]) = [0, 1).\)
• If one considers on \(\Reals\) the topology in which every set is open, then \(\operatorname{int} ([0, 1]) = [0, 1].\)
• If one considers on \(\Reals\) the topology in which the only open sets are the empty set and \(\Reals\) itself, then \(\operatorname{int} ([0, 1])\) is the empty set.

These examples show that the interior of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.

• In any discrete space, since every set is open, every set is equal to its interior.
• In any indiscrete space \(X,\) since the only open sets are the empty set and \(X\) itself, \(\operatorname{int} X = X\) and for every proper subset \(S\) of \(X,\) \(\operatorname{int} S\) is the empty set.

Let \(X\) be a topological space and let \(S\) and \(T\) be subsets of \(X.\)

• \(\operatorname{int} S\) is open in \(X.\)
• If \(T\) is open in \(X\) then \(T \subseteq S\) if and only if \(T \subseteq \operatorname{int} S.\)
• \(\operatorname{int} S\) is an open subset of \(S\) when \(S\) is given the subspace topology.
• \(S\) is an open subset of \(X\) if and only if \(\operatorname{int} S = S.\)
• Intensive: \(\operatorname{int} S \subseteq S.\)
• Idempotence: \(\operatorname{int} (\operatorname{int} S) = \operatorname{int} S.\)
• Preserves/distributes over binary intersection: \(\operatorname{int} (S \cap T) = (\operatorname{int} S) \cap (\operatorname{int} T).\)
  • However, the interior operator does not distribute over unions since only \(\operatorname{int} (S \cup T) ~\supseteq~ (\operatorname{int} S) \cup (\operatorname{int} T)\) is guaranteed in general and equality might not hold. For example, if \(X = \Reals, S = (-\infty, 0],\) and \(T = (0, \infty)\) then \((\operatorname{int} S) \cup (\operatorname{int} T) = (-\infty, 0) \cup (0, \infty) = \Reals \setminus \{0\}\) is a proper subset of \(\operatorname{int} (S \cup T) = \operatorname{int} \Reals = \Reals.\)
• Monotone/nondecreasing with respect to \(\subseteq\): If \(S \subseteq T\) then \(\operatorname{int} S \subseteq \operatorname{int} T.\)

Other properties include:

• If \(S\) is closed in \(X\) and \(\operatorname{int} T = \varnothing\) then \(\operatorname{int} (S \cup T) = \operatorname{int} S.\)

Relationship with closure

The above statements will remain true if all instances of the symbols/words

"interior", "int", "open", "subset", and "largest"

are respectively replaced by

"closure", "cl", "closed", "superset", and "smallest"

and the following symbols are swapped:

1. "\(\subseteq\)" swapped with "\(\supseteq\)"
2. "\(\cup\)" swapped with "\(\cap\)"

For more details on this matter, see interior operator below or the article Kuratowski closure axioms.

The interior operator \(\operatorname{int}_X\) is dual to the closure operator, which is denoted by \(\operatorname{cl}_X\) or by an overline , in the sense that \[\operatorname{int}_X S = X \setminus \overline{(X \setminus S)}\] and also \[\overline{S} = X \setminus \operatorname{int}_X (X \setminus S),\] where \(X\) is the topological space containing \(S,\) and the backslash \(\,\setminus\,\) denotes set-theoretic difference. Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators, by replacing sets with their complements in \(X.\)

In general, the interior operator does not commute with unions. However, in a complete metric space the following result does hold:

Theorem (C. Ursescu), Let \(S_1, S_2, \ldots\) be a sequence of subsets of a complete metric space \(X.\)

• If each \(S_i\) is closed in \(X\) then \[{\operatorname{cl}_X} \biggl( \bigcup_{i \in \N} \operatorname{int}_X S_i \biggr) = {\operatorname{cl}_X \operatorname{int}_X} \biggl( \bigcup_{i \in \N} S_i \biggr).\]
• If each \(S_i\) is open in \(X\) then \[{\operatorname{int}_X} \biggl( \bigcap_{i \in \N} \operatorname{cl}_X S_i \biggr) = {\operatorname{int}_X \operatorname{cl}_X} \biggl( \bigcap_{i \in \N} S_i \biggr).\]

The result above implies that every complete metric space is a Baire space.

The exterior of a subset \(S\) of a topological space \(X,\) denoted by \(\operatorname{ext}_X S\) or simply \(\operatorname{ext} S,\) is the largest open set disjoint from \(S,\) namely, it is the union of all open sets in \(X\) that are disjoint from \(S.\) The exterior is the interior of the complement, which is the same as the complement of the closure; in formulas, \[\operatorname{ext} S = \operatorname{int}(X \setminus S) = X \setminus \overline{S}.\]

Similarly, the interior is the exterior of the complement: \[\operatorname{int} S = \operatorname{ext}(X \setminus S).\]

The interior, boundary, and exterior of a set \(S\) together partition the whole space into three blocks (or fewer when one or more of these is empty): \[X = \operatorname{int} S \cup \partial S \cup \operatorname{ext} S,\] where \(\partial S\) denotes the boundary of \(S.\) The interior and exterior are always open, while the boundary is closed.

Some of the properties of the exterior operator are unlike those of the interior operator:

• The exterior operator reverses inclusions; if \(S \subseteq T,\) then \(\operatorname{ext} T \subseteq \operatorname{ext} S.\)
• The exterior operator is not idempotent. It does have the property that \(\operatorname{int} S \subseteq \operatorname{ext}\left(\operatorname{ext} S\right).\)

Two shapes \(a\) and \(b\) are called interior-disjoint if the intersection of their interiors is empty. Interior-disjoint shapes may or may not intersect in their boundary.