Notes on Topology

Let $X$ be a set, and let $F$ be an ordered field. An $F$-valued pseudometric on $X$ is a function $d: X \times X \to F$ that satisfies the following requirements:

• $d(x, x) = 0_F$ for all $x \in X$.
• Symmetry: $d(x, y) = d(y, x)$ for all $x, y \in X$.
• Triangle inequality: $d(x, z) \le d(x, y) + d(y, z)$ for all $x, y, z \in X$.

Theorem: Let $X$ be a set, let $F$ be an ordered field, and let $d: X \times X \to F$ be an $F$-valued pseudometric on $X$. For all $x, y \in X$, we have $d(x, y) \ge 0_F$.

Proof: Let $x, y \in X$ be given. By the defining properties of a pseudometric, we have

which implies that $d(x, y) \ge 0_F$. (Note that $2_F \ne 0_F$ in all ordered fields $F$.) ∎

Let $X$ be a set, and let $F$ be an ordered field. An $F$-valued metric on $X$, also known as an $F$-valued distance function on $X$, is an $F$-valued pseudometric $d: X \times X \to F$ that satisfies the following additional requirement:

• Positivity: For all $x, y \in X$, if $d(x, y) = 0_F$, then $x = y$.

A generalized pseudometric space is an ordered triple $(X, F, d)$ consisting of a set $X$, called the underlying set, an ordered field $F$, called the distance field, and an $F$-valued pseudometric $d$ on $X$. The elements of $X$ are called points.

Let $(X, F, d)$ be a generalized pseudometric space, $x \in X$, and $r \in F$. The open ball of radius $r$ centered at $x$, denoted by $B_r(x)$, is the subset of $X$ defined by

Similarly, the closed ball of radius $r$ centered at $x$, denoted by $B_r[x]$, is the subset of $X$ defined by

Let $(X, F_X, d_X)$ and $(Y, F_Y, d_Y)$ be generalized pseudometric spaces. A function $f: X \to Y$ is continuous at a point $x_0 \in X$ if, for every $\varepsilon > 0_{F_Y}$, there exists $\delta > 0_{F_X}$ such that $f[B_\delta(x_0)] \subseteq B_\varepsilon(f(x_0))$.

We say that $f$ is everywhere continuous, or simply continuous, if $f$ is continuous at every point $x_0 \in X$.

Let $(X, F, d)$ be a generalized pseudometric space, and let $A \subseteq X$. A point $x \in X$ is an interior point of $A$ if there exists $r > 0_F$ such that $B_r(x) \subseteq A$. The set of all interior points of $A$, denoted by $A^\circ$, is called the interior of $A$.

We say that $A$ is open if every point in $A$ is an interior point of $A$ (i.e., $A \subseteq A^\circ$).

Let $(X, F, d)$ be a generalized pseudometric space, and let $A \subseteq X$. A point $x \in X$ is close to $A$, or touches $A$, if for every $r > 0_F$, there exists $a \in A$ such that $d(x, a) < r$. The set of all points close to $A$, denoted by $\overline{A}$, is called the closure of $A$.

We say that $A$ is closed if $A$ contains every point that is close to $A$ (i.e., $\overline{A} \subseteq A$).

Let $X$ be a set. A topology on $X$ is a collection $T \subseteq \powerset{X}$ of subsets of $X$ that satisfies the following requirements:

• Empty set is open: $\varnothing \in T$.
• Whole set is open: $X \in T$.
• Closed under arbitrary unions: If $S \subseteq T$, then $\bigcup S \in T$.
• Closed under finite intersections: If $S \subseteq T$ and $\abs{S}$ is finite, then $\bigcap S \in T$.

A topological space is an ordered pair $(X, T)$ consisting of a set $X$, called the underlying set, and a topology $T$ on $X$. The elements of $X$ are called the points of the topological space $(X, T)$, and the elements of $T$ are called its open sets.

Let $(X, T)$ be a topological space. A subset $F \subseteq X$ is closed if $X \setminus F \in T$.

Let $(X, T)$ be a topological space. A subset of $X$ is clopen if it is both open and closed.