The Timpson Lyceum · General Topology (Point Set Topology)

Functions & Open Sets in a Topology – Lecture 5 (Topology)

Overview

This lecture builds the language of functions and then uses it to reach the central definition of point-set topology: the topology itself, a collection of "open sets." We first sharpen the chain set → relation → function, review the key properties a function can have (one-to-one, onto, bijection) and how functions compose, and use bijections to make precise what it means for a set to be finite, countably infinite, or uncountable. With that vocabulary in hand, we define a topological space: a set together with a chosen collection of subsets, the open sets, satisfying three axioms. Topology matters because it gives a setting general enough to talk about continuity even when "don't lift your pencil" no longer makes sense — and the familiar calculus of the real line turns out to be just one special topology.

Key Concepts

Sets, relations, functions (the hierarchy).

• A set is an unordered collection with no repeats: $\{a,b,c\} = \{c,b,a\} = \{a,a,b,c,c\}$, all the same three-element set.
• A relation is a set whose elements are ordered pairs. Order matters in a pair: $(a,b) \ne (b,a)$, just as the point $(1,5)$ differs from $(5,1)$.
• A function is a relation in which every left-hand element is paired with a unique right-hand element. So $\{(a,b),(a,c)\}$ is not a function ($a$ goes to two things), but $\{(a,b),(c,b)\}$ is.

Function notation.

• For a function $f$, $f(3)$ denotes the unique right-hand element paired with $3$. The notation only makes sense when $f$ is genuinely a function.
• $f : X \to Y$ reads "$f$ is a function from $X$ to $Y$." The domain is all of $X$; the image (also called range) is the set of right-hand elements, which is a subset of $Y$. $Y$ itself is the codomain.
• Example: $f : \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$. The domain is all of $\mathbb{R}$, but the image is only $[0,\infty)$ — you cannot produce a negative output.

Properties a function can have.

• One-to-one (injective): distinct inputs give distinct outputs. Formally, $f$ is one-to-one iff $f(x)=f(y) \implies x=y$. Graphically, it passes the horizontal-line test. $x^2$ is not one-to-one ($f(1)=f(-1)$); $x^3$ is.
• Onto (surjective): the image is all of $Y$, i.e. $f(X) = Y$. A cubic $\mathbb{R}\to\mathbb{R}$ is onto; a parabola $x^2$ is not (it misses the negatives).
• Bijection: a function that is both one-to-one and onto.

Function composition.

• Given $f : X \to Y$ and $g : Y \to Z$, the composition $g \circ f : X \to Z$ is defined by $(g \circ f)(x) = g(f(x))$ — feed $x$ through $f$, then through $g$. The codomain $Y$ of $f$ must match the domain of $g$; that middle set is "glued out."
• Composition is not commutative: in general $g \circ f \ne f \circ g$.

Sizes of sets via bijection. Two sets have the same size (cardinality) exactly when a bijection pairs them off perfectly.

• Finite: $X$ is finite if it is empty, or there is a bijection $f : \{0,1,2,\dots,n\} \to X$ for some $n \in \mathbb{Z}^+$. (Counting is building such a bijection.) Not finite means infinite.
• Countably infinite: there is a bijection between $X$ and the naturals $\mathbb{N}$. The integers $\mathbb{Z}$ and the rationals $\mathbb{Q}$ are both countably infinite — same size as $\mathbb{N}$, despite intuition.
• Uncountable: infinite but with no bijection to $\mathbb{N}$. The reals $\mathbb{R}$ are uncountable — a strictly larger infinity. So $|\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}| < |\mathbb{R}|$.
• Useful closure facts: a subset of a finite (countable) set is finite (countable); finite unions/products of finite sets are finite; countable unions/products of countable sets are countable.

Sequences. A sequence in $X$ is a function $f : \mathbb{Z}^+ \to X$; we write it by listing outputs, e.g. $f(n)=n^2$ as $1,4,9,16,\dots$ A subsequence picks out infinitely many terms while preserving their original order, indexed by a strictly increasing sequence $k_1 < k_2 < k_3 < \cdots$ of positions, so $y_n = x_{k_n}$.

Topology (the main definition). Let $X$ be a set. A topology $\tau$ on $X$ is a collection of subsets of $X$ — its open sets — such that:

1. $\varnothing \in \tau$ and $X \in \tau$.
2. The intersection of finitely many open sets is open.
3. The union of any collection of open sets is open.

A set with a chosen topology is a topological space. "Open" means nothing more than "is a member of $\tau$."

Standard examples of topologies.

• Trivial (indiscrete) topology: $\tau = \{\varnothing, X\}$ — the smallest possible.
• Discrete topology: $\tau = \mathcal{P}(X)$, the power set (every subset is open) — the largest possible.
• There is no single "right" topology on a set; many choices satisfy the axioms, and the interesting ones lie between trivial and discrete. (Continuity in topology is defined by "pre-images of open sets are open," which on $\mathbb{R}$ recovers the calculus $\varepsilon$–$\delta$ definition.)

Worked Example

Is $\tau$ a topology? Let $X=\{a,b,c\}$ and test a candidate collection against the three axioms.

The union axiom fails, so this is not a topology — adding $\{a,b\}$ to the collection repairs it.

By contrast, a nested family always works:

Every intersection and union lands back inside $\tau$, and both $\varnothing$ and $X$ are present, so this is a topology.

Summary

• The structures nest: every function is a special relation, and every relation is a special set; $f:X\to Y$ has domain all of $X$ and image a subset of the codomain $Y$.
• A function is one-to-one when distinct inputs give distinct outputs, onto when its image is all of $Y$, and a bijection when both; composition $(g\circ f)(x)=g(f(x))$ chains two functions and is generally not commutative.
• Bijections measure size: finite sets pair with $\{0,\dots,n\}$; countably infinite sets pair with $\mathbb{N}$ (true of $\mathbb{Z}$ and $\mathbb{Q}$); $\mathbb{R}$ is uncountable, a strictly bigger infinity.
• A topology $\tau$ on $X$ is a family of subsets (the open sets) closed under finite intersection and arbitrary union, containing $\varnothing$ and $X$; checking it means verifying these axioms, not trusting intuition.
• The trivial topology $\{\varnothing,X\}$ and the discrete topology $\mathcal{P}(X)$ are the extreme cases; a set generally admits many valid topologies, and topology exists to talk about continuity in this fully general setting.