The Timpson Lyceum · General Topology (Point Set Topology)

Axiomatic Set Theory – Lecture 1 (Topology)

Axiomatic Set Theory – Lecture 1

Overview

This lecture opens the topology course by building its foundation: axiomatic set theory. Topology is introduced as the study of proximity — the structural relation it preserves (homeomorphism) cares only about which points are "close," not about distances, just as a rubber band can be stretched without separating neighboring points. Because every mathematical object will ultimately be defined as a set, we first lay down a rigorous, axiom-based theory of sets to avoid the contradictions of naive set theory. The lecture covers the first several of nine axioms, the core definitions (subset, equality, ordered pair), and the construction of set-builder notation.

Key Concepts

Topology as the study of proximity. Every field of mathematics has its own "sameness" relation: congruence in geometry, isomorphism in algebra. In topology the sameness relation is homeomorphism — two spaces are the same if one can be deformed into the other (stretched like a rubber band, without tearing or gluing) so that proximity is preserved. Distances may change freely; only closeness matters. The familiar notion of a continuous function from $\mathbb{R}$ to $\mathbb{R}$ is just one special case (the standard topology on $\mathbb{R}$) of this more general idea.

Primitives. The theory leaves two notions undefined:

• set — the only kind of object in the theory (sets contain only other sets, "all the way down").
• membership $\in$ — the relation "$x$ is a member of $y$." It is a relation: it takes two terms and returns true or false.

Axiom 1 (membership is a proposition). The expression $x \in y$ is a proposition (true or false, never both) if and only if $x$ and $y$ are both sets: $\forall x\, \forall y \,\big( (x \in y) \veebar \neg(x \in y) \big)$ This rules out defining objects that aren't sets. It is motivated by Russell's Paradox: let $U = \{\, x : x \notin x \,\}$, the set of all sets that don't contain themselves. Asking whether $U \in U$ gives $U \in U \iff U \notin U$ — a contradiction. The resolution: "$U \in U$" is neither true nor false, so it isn't a proposition, so by Axiom 1, $U$ is not a set. You cannot conjure a set into existence merely by writing it down.

Basic definitions (all built from $\in$):

• Not a member: $x \notin y \iff \neg(x \in y)$.
• Subset: $X \subseteq Y \iff \forall a\,(a \in X \rightarrow a \in Y)$.
• Set equality: $X = Y \iff (X \subseteq Y \;\wedge\; Y \subseteq X)$. This extends logical identity so that order and repetition don't matter, e.g. $\{1,2\} = \{2,1\}$.
• Proper subset: $X \subsetneq Y \iff (X \subseteq Y \;\wedge\; X \neq Y)$. No set is a proper subset of itself.

Axiom 2 (empty set exists). $\exists x\, \forall y\,(y \notin x)$. A theorem then proves this set is unique, licensing the definition of the constant $\varnothing$, the empty set.

Axiom 3 (pairing). If $X$ and $Y$ are sets, then $\{X, Y\}$ — the set containing precisely $X$ and $Y$ — is a set: $\forall X\, \forall Y\, \exists M\, \forall u\,\big(u \in M \iff (u = X \,\vee\, u = Y)\big)$ This is what introduces the brace notation. A singleton is defined as $\{X\} := \{X, X\}$. So far only one- and two-element sets are defined.

Ordered pairs as sets. Order is not built in, so it must be encoded: $(X, Y) := \{\, \{X\},\; \{X, Y\} \,\}$ The left element is recoverable (it is the one belonging to both inner sets / the member of the other), so the information is preserved: $(X,Y) = (a,b) \iff X = a \wedge Y = b$. Triples and longer tuples are defined recursively, e.g. $(X, Y, Z) := \big((X, Y),\, Z\big)$.

Axiom 4 (union). If $X$ is a set, then $\bigcup X$ — whose elements are the elements of the elements of $X$ — is a set: $\forall X\, \exists M\, \forall y\,\big(y \in M \iff \exists u\,(u \in X \wedge y \in u)\big)$ This is used to define the familiar multi-element brace notation: $\{a,b,c\} := \bigcup\{\,\{a,b\},\{c\}\,\}$, extended recursively to $\{a_1, \dots, a_n\}$ for $n \ge 3$.

Functional relations. A relation $R$ is functional if every input is related to exactly one output: $\forall x\, \exists!\, y\,(x \,R\, y)$. Example: the doubling relation, where $x$ maps to $2x$.

Axiom 5 (replacement / image). If $M$ is a set and $R$ is a functional relation, then the image $\mathrm{Im}(R, M)$ — all $y$ such that some $x \in M$ has $x \,R\, y$ — is a set.

Set-builder notation (theorem from Axiom 5). For a predicate $P$ and a set $M$, the collection $\{\, y \in M : P(y) \,\}$ is a set. Crucially, the element must be drawn from an existing set $M$ — you can never write $\{\, y : P(y)\,\}$ unrestricted, which is exactly the move that produced Russell's Paradox.

Worked Example

Claim (pairing is unordered): $\{X, Y\} = \{Y, X\}$.

To prove two sets equal, show each is a subset of the other (set equality), and to show $A \subseteq B$, take an arbitrary $a \in A$ and prove $a \in B$.

Every step is a logical equivalence ($\iff$), so the containment holds in both directions and the two sets are equal — confirming that listing order does not distinguish a set.

Summary

• Topology studies proximity, not distance; its sameness relation is homeomorphism, and ordinary continuity on $\mathbb{R}$ is just the standard topology, a special case.
• The theory has exactly two primitives — set and membership $\in$ — and everything else is built from them; every mathematical object is ultimately a set.
• Axiom 1 makes $x \in y$ a proposition only when both are sets, blocking Russell's Paradox ($U = \{x : x \notin x\}$ is not a set).
• Core definitions: subset ($\subseteq$), equality (mutual containment, so $\{1,2\}=\{2,1\}$), and proper subset ($\subsetneq$).
• Axioms 2–4 give the empty set (provably unique), pairing $\{X,Y\}$, and union $\bigcup X$; ordered pairs are encoded as $(X,Y) = \{\{X\},\{X,Y\}\}$ and tuples defined recursively.
• Axiom 5 (image of a functional relation is a set) yields set-builder notation $\{y \in M : P(y)\}$, which must always draw from an existing set — the safeguard that naive set theory lacked.