1 point-set topology in \mathbb R^n

Working in real euclidean space:

• \mathbb R^n = \{(x_1, \cdots, x_n) : x_i \in \mathbb R\}
• ||x - y|| = \sqrt{(x_1 - y_1)^2 + \cdots + (x_n - y_n)^2}
• D^n(x, r) = \{y \in \mathbb R^n : ||x - y|| < r\}

1.1 point classification

Let A \subseteq \mathbb R^n

1. x is an interior point of A if there is an open ball B such that x \in B \subseteq A
2. x is an exterior point of A if there is an open ball B such that x \in B \subseteq (\mathbb R^n \setminus A)
3. x is a limit point of A if every ball B around x contains point of A: \forall_{r > 0} B^n(x, r) \cap A \neq \emptyset
4. x is a frontier point of A if every ball B around x contains both points of A and points of \mathbb R^n \setminus A: \forall_{r > 0} (B^n(x, r) \cap A \neq \emptyset \land B^n(x, r) \cap (\mathbb R^n \setminus A) \neq \emptyset)

A point is always one of the 1., 2., or 4., and always only one of them.

1.2 set properties

Let A \subseteq \mathbb R^n

• a set A is open if every x \in A is an interior point
• a set A is closed if every x \notin A is an exterior point
• a set A is bounded if A \subseteq D^n(0, r) for some r

1.2.1 set functions

• interior of a set A denoted by Int(A) is the set of all interior points of A
• frontier of a set A denoted by Fr(A) is the set of all frontier points of A
• closure of a set A denoted by Cl(A) is the set A \cup Fr(A)

1.3 limit points of sequences

A point p \in \mathbb R^n is a limit point of a sequence \{x_i\}_{i=1}^\infty if every neighborhood of p contains an infinite number of x_i.

If p is a limit point of a set A \subseteq \mathbb R^n then there exists a sequence of points \{x_i\}_{i=1}^\infty where x_i \in A such that p is a limit point of the sequence.

If \{x_i\}_{i=1}^\infty is a sequence with each x_i \in A and p is a limit point of the sequence then p is a limit point of A.

1.4 relative neighborhoods

A neighborhood of a point x \in A relative to A is a set of the form D^n(x, r) \cap A.

Let B \subseteq A. The set B is open relative to A if every x \in B is an interior point relative to A.

Let B \subseteq A \subseteq \mathbb R^n:

• B is open relative to A iff B = A \cap U for some open set U \subseteq \mathbb R^n
• B is closed relative to A iff B = A \cap C for some closed set C \subseteq \mathbb R^n

1.5 continuity

Let D \subseteq \mathbb R^n and R \subseteq \mathbb R^m.

• A function f: D \to R is continuous if whenever U is open in R, the set f^{-1}(U) is an open set in D
• A function f: D \to R is continuous iff x is a limit point of a set B \subseteq D then f(x) is a limit point of f(B) in R
• A function f: D \to R is continuous iff for every x \in D and every \varepsilon > 0 there exists a \delta > 0 such that if y \in D and ||y - x|| < \delta then ||f(y) - f(x)|| < \varepsilon

A function f: X \to Y is a homeomorphism if f is continuous and f^{-1} is continuous. The spaces X and Y are called topologically equivalent (homeomorphic).

1.6 topological property

P is a topological property if whenever set A has property P and set B is topologically equivalent to A, then B has property P.

1.7 compactness

A subset A of \mathbb R^n is compact if every sequence of points in A has a convergent subsequence to a point in A.

Let A \subseteq \mathbb R^n be a compact subset, and let f: A \to \mathbb R^m be a continuous map. Then f(A) is a compact set.

1.7.1 Heine-Borel theorem

A subset A of \mathbb R^n is compact iff it is closed and bounded.

1.8 connectedness

A set S is connected if whenever S is divided into two nonempty disjoint sets, one contains a limit point of the other.

The set S is connected iff S cannot be written as a union of two nonempty disjoint sets which are open relative to S.

If f: D \to R is a continuous function from a connected set D onto a set R, then R is connected.

[0; 1] is compact and connected.

1.9 fixed point

If f: X \to X a point x_0 \in X such that f(x_0) = x_0 is called a fixed point of f

If f: [0; 1] \to [0; 1] is a continuous function, then there is a fixed point of f, ie a point x_0 \in [0; 1] such that f(x_0) = x_0

A space X has the fixed point property if every continuous map f: X \to X has a fixed point.

The fixed point property is a topological property.

1.10 antipodes

The points on an n-sphere x = (x_1, \cdots, x_n) and -x = (-x_1, \cdots, -x_n) are called antipodal.

1.10.1 Borsuk-Ulam

If f: S^1 \to \mathbb R is continuous, then there exists an x_0 \in S^1 such that f(x_0) = f(-x_0).

At any one point there are two diametrically opposite points with the same temperature.

1.11 sandwich theorem

Let A, B be bounded connected open subsets in \mathbb R^2 which may overlap. There exists a line which divides each region in half.