Introduction

 

• What topology is. From Greek topos (place) and logos (discourse, reason): the study of continuous functions, also called maps.

• The problem. To make sense of “$f:X\to Y$ is continuous” we need extra data: continuity says that if $x,y\in X$ are close, then $f(x),f(y)$ are close. So $X$ and $Y$ each need a notion of closeness.

• First answer: metrics. Assign a distance $d(x,y)$; call $x,y$ close when $d(x,y)$ is small. This gives a metric space $(X,d)$.

  Note. A bare set carries no information about two elements beyond whether they are equal. A metric gives it a shape — which is why we then say space rather than set, and point rather than element.

• Why that is not enough. Metric spaces are special. Pointwise convergence of real functions ($f_n\to g$ iff $f_n(t)\to g(t)$ for each $t$) is a useful notion of closeness, but no metric on the set of real functions expresses it.

• $\mathrm{★}$ Second answer: open sets. Instead of distances between points, specify which subsets $U\subset X$ are open. Read this as:

  • if $x$ lies in $U$ and $U$ is open, then every $y$ sufficiently close to $x$ also lies in $U$.

  The collection of all open subsets is the topology $𝒯$ on $X$.

• Consequence for the course. This approach handles not just elements and functions but subsets and collections of subsets — hence we begin with a summary of set theory.

• $\mathrm{★}$ The payoff. For topological spaces the definition of continuity is simply:

  • for each open $V\subset Y$, the preimage $f^{-1}(V)$ is open in $X$.

  Compare the $(\epsilon ,\delta )$ definition for metric spaces: for each $x\in X$ and each $\epsilon >0$ there is $\delta >0$ such that implies .

• On the abstraction. The definition of a topological space looks harder than the subsequent definition of a continuous map. Same pattern as linear algebra: “$f$ is linear if $f(\lambda x+\mu y)=\lambda f(x)+\mu f(y)$” is simple, but presupposes the abstract definition of a real vector space, which in turn presupposes the nine-odd field axioms for $ℝ$.

  Note. Moral: axiomatisations of the most fundamental objects are general enough to be hard to grasp at once. It is the relations between them — continuous maps, linear transformations — that are concrete. Say this early; it defuses a lot of anxiety.

• Programme. After spaces and maps we study the properties such spaces may have:

  • connected — not a disjoint union of subspaces;

  • compact — not too many open subsets globally;

  • Hausdorff — enough open subsets locally.

  Then the consequences: general forms of the intermediate value theorem, existence of maximal values, uniqueness of limits, and more.

• Notation. These notes follow J. R. Munkres, Topology; the $\mathrm{\S }$-signs refer to sections of that book. Course MAT3500/4500, University of Oslo.

Where this leads (worth 5 minutes at the first lecture).

• Classification of surfaces. Two facts determine a connected compact $2$-manifold up to topological equivalence: whether it can be oriented, and how many handles it has. The number of handles is the genus $g$: sphere $g=0$, torus $g=1$, two-handled surface $g=2$.

• Gauss–Bonnet — local geometry against global topology. For a surface $F$ with a Riemannian metric,

  $${\int }_{F}K𝑑A=\mathrm{\hspace{0.33em}2}\pi \cdot \chi ,\chi =2-2g.$$

  Check on the sphere of radius $r$: curvature $1/r^2$ everywhere, area $4\pi r^2$, product $4\pi =2\pi \cdot 2$, and indeed $\chi (S^2)=2$. [Mk course MAT4510]

• Topology and rational points. For curves over $ℂ$: $x^2+y^2=1$ is a sphere and has infinitely many rational solutions; $x^5+y^2=1$ has genus $2$ and only finitely many. Conjectured by Mordell, proved by Faltings (1983): a rationally defined algebraic curve of genus $>1$ has only finitely many rational points.

  Note. A topological condition forcing an arithmetic conclusion. Good advertisement for the subject. [Mk courses in algebraic geometry]

1  Set Theory and Logic

1.1  (§1) Fundamental concepts

• Notation. $x\in A$, $x\notin A$. $D=\{0,1,\mathrm{\dots },9\}$,  $P=\{n\in \mathbb{N}\mid n\text{ prime}\}$,  $S=\{n^2\mid n\in \mathbb{N}\}$.

• Caution. $x\ne \{x\}$. $\mathrm{\varnothing }=\{\}$ has no elements.

• Def. $A\subset B$ iff $(x\in A)\Rightarrow (x\in B)$. $A=B$ iff $A\subset B$ and $B\subset A$. $A⊊B$: proper.

• Def. $A\cap B=\{x\mid x\in A\text{ and }x\in B\}$, $A\cup B=\{x\mid x\in A\text{ or }x\in B\}$ (inclusive or).

• Distributive laws. $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$, $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$.

• Def. $A-B=\{x\in A\mid x\notin B\}$ — the complement of $B$ in $A$.

• $\mathrm{★}$ De Morgan. $A-(B\cup C)=(A-B)\cap (A-C)$, $A-(B\cap C)=(A-B)\cup (A-C)$.

  Note. Used constantly later: it is what converts the union/intersection axioms for open sets into the intersection/union statements for closed sets.

• Notation. Informal listings such as $P=\{2,3,5,\mathrm{\dots }\}$ and $S=\{1,4,9,\mathrm{\dots }\}$ are used when the pattern is clear from context.

• Def. A set whose elements are sets is called a collection (or family), written $𝒜,ℬ,\mathrm{\dots }$ For a given $A$, the power set is $𝒫(A)=\{B\mid B\subset A\}$. E.g. $𝒫(\{a,b\})=\{\mathrm{\varnothing },\{a\},\{b\},\{a,b\}\}$, with four elements.

• Ex. (Students and courses — a collection to keep in mind.) $S=\{s\mid s$ a student$\}$, $C=\{c\mid c$ a course$\}$, $E_c=\{s\in S\mid s$ enrolled in $c\}$, and $ℰ=\{E_c\mid c\in C\}$. Elements of $ℰ$ are sets of students.

  Note. Note $E_c=E_d$ can happen for $c\ne d$ (both empty). Worth pointing out before indexed families in §5, where exactly this failure of injectivity is allowed.

• Def. For a collection $𝒜$: ${\bigcap }_{A\in 𝒜}A=\{x\mid x\in A\forall A\in 𝒜\}$, ${\bigcup }_{A\in 𝒜}A=\{x\mid x\in A\text{ for some }A\in 𝒜\}$.

• Caution. ${\bigcup }_{A\in \mathrm{\varnothing }}A=\mathrm{\varnothing }$, but ${\bigcap }_{A\in \mathrm{\varnothing }}A$ needs an ambient universal set $U$; then it is $U$.

• Ex. With that collection: ${\bigcap }_{E_c\in ℰ}{E}_c$ = students enrolled in every course (probably empty); ${\bigcup }_{E_c\in ℰ}{E}_c$ = the active students; $S-\bigcup E_c$ = the inactive ones.

• Def. $A\times B=\{(x,y)\mid x\in A,y\in B\}$; $(x,y)=(x^{\prime },y^{\prime })$ iff $x=x^{\prime }$ and $y=y^{\prime }$. Distinguish the ordered pair $(x,y)$ from the set $\{x,y\}$. (If wanted: define $(x,y)=\{\{x\},\{x,y\}\}$.)

  Note. ${ℝ}^2=ℝ\times ℝ$ read as horizontal and vertical coordinates is Descartes’ analytic geometry, as against Euclid’s synthetic approach.

1.2  (§2) Functions

• Def. $f:A\to B$: to each $x\in A$ a unique $f(x)\in B$. $A$ = domain, $B$ = range (codomain).

• Ex. The rule may be given by a procedure, e.g. for $x\in \mathbb{N}$

  but no such assumption is made in general.

• Def. Graph ${\mathrm{\Gamma }}_f=\{(x,f(x))\}\subset A\times B$. Characterised by: for each $x\in A$ exactly one $y$ with $(x,y)\in \mathrm{\Gamma }$.

  Note. So a function may be defined as a triple $(A,B,\mathrm{\Gamma })$ — domain and codomain are part of the data, not just the rule.

• Def. Image $f(A)=\{f(x)\mid x\in A\}\subset B$. Restriction $f|S:S\to B$. Corestriction $A\to T$, defined only when $f(A)\subset T$.

• Def. injective: $f(x)=f(y)\Rightarrow x=y$. surjective: $f(A)=B$. bijective: both; then $f^{-1}:B\to A$ exists, with $f^{-1}(y)=x$ exactly when $y=f(x)$.

• Rmk. ${\mathrm{\Gamma }}_{f^{-1}}=\{(y,x)\in B\times A\mid (x,y)\in {\mathrm{\Gamma }}_f\}$: the graph of $f^{-1}$ is the graph of $f$ with the two factors interchanged.

• Rmk. If $f|S$ is injective and $T=f(S)$, the resulting $g:S\to T$ is a bijection with an inverse $g^{-1}:T\to S$, even when $f$ itself is not invertible.

• Def. $(g\circ f)(x)=g(f(x))$. Unital, associative, not commutative. ${(g\circ f)}^{-1}=f^{-1}\circ g^{-1}$.

• Def. Inclusion $i:S\to A$, $i(x)=x$ for $x\in S$. Not the identity unless $S=A$. Then $f\circ i=f|S$.

• Rmk. Dually, for $T\subset B$ with $f(A)\subset T$ and $j:T\to B$ the inclusion, the corestriction $g:A\to T$ is characterised by $j\circ g=f$.

• Def. Induced maps on power sets: $f:𝒫(A)\to 𝒫(B)$, $f^{-1}:𝒫(B)\to 𝒫(A)$.

  Note. Same symbol, different domain — flag the abuse of notation once and move on.

Images vs. preimages — the asymmetry to put on the board:

• Images (only partly well behaved). For $S,T\subset A$:

  • $S\subset T\Rightarrow f(S)\subset f(T)$

  • $f(S\cup T)=f(S)\cup f(T)$

  • $f(S\cap T)\subset f(S)\cap f(T)$  — equality if $f$ injective

  • $f(T)-f(S)\subset f(T-S)$  — equality if $f$ injective

• $\mathrm{★}$ Preimages (all four are equalities). For $S,T\subset B$:

  • $S\subset T\Rightarrow f^{-1}(S)\subset f^{-1}(T)$

  • $f^{-1}(S\cup T)=f^{-1}(S)\cup f^{-1}(T)$

  • $f^{-1}(S\cap T)=f^{-1}(S)\cap f^{-1}(T)$

  • $f^{-1}(S-T)=f^{-1}(S)-f^{-1}(T)$

• Round trips. $S\subset f^{-1}(f(S))$ and $f(f^{-1}(T))\subset T$.

  Note. This table is the whole reason continuity will be defined by preimages, not images. Point forward to it now; refer back in §18.