1  Topological Spaces and Continuous Functions

1.1  (§12) Topological spaces

• $\mathrm{★}$ Def. A topology on a set $X$ is a collection $𝒯$ of subsets with

  1. $\mathrm{\varnothing }\in 𝒯$ and $X\in 𝒯$;

  2. ${\{U_{\alpha }\}}_{\alpha \in J}\subset 𝒯\Rightarrow {\bigcup }_{\alpha \in J}{U}_{\alpha }\in 𝒯$ (arbitrary unions)

  3. $U_1,\mathrm{\dots },U_n\in 𝒯\Rightarrow U_1\cap \mathrm{\cdots }\cap U_n\in 𝒯$ (finite intersections)

  $(X,𝒯)$ is a topological space; the $U\in 𝒯$ are the open sets.

  Note. For (3) it is enough to check $n=2$; induct. The asymmetry between (2) and (3) is the whole subject — infinite intersections of open sets need not be open: ${\bigcap }_n(-1/n,1/n)=\{0\}$.

• Def. Discrete ${𝒯}_{\mathrm{disc}}=𝒫(X)$ (every subset open). Trivial ${𝒯}_{\mathrm{triv}}=\{\mathrm{\varnothing },X\}$.

• Ex. $X=\{a,b\}$: exactly four topologies — ${𝒯}_{\mathrm{triv}}$,   ${𝒯}_a=\{\mathrm{\varnothing },\{a\},X\}$,   ${𝒯}_b=\{\mathrm{\varnothing },\{b\},X\}$,   ${𝒯}_{\mathrm{disc}}$. The middle two are the Sierpiński topologies.

  Note. In $(X,{𝒯}_a)$: $a$ is separated from $b$, but every open set containing $b$ contains $a$. So $a$ is “arbitrarily close” to $b$ without $b$ being close to $a$. No metric does this.

• Ex. $X=\{a,b,c\}$: there are $29$ topologies. Nine of them, the rest obtained by permuting $a,b,c$:

  • $\{\mathrm{\varnothing },X\}$

  • $\{\mathrm{\varnothing },\{a\},X\}$

  • $\{\mathrm{\varnothing },\{a,b\},X\}$

  • $\{\mathrm{\varnothing },\{a\},\{a,b\},X\}$

  • $\{\mathrm{\varnothing },\{a,b\},\{c\},X\}$

  • $\{\mathrm{\varnothing },\{a\},\{b\},\{a,b\},X\}$

  • $\{\mathrm{\varnothing },\{a\},\{a,b\},\{a,c\},X\}$

  • $\{\mathrm{\varnothing },\{a\},\{c\},\{a,b\},\{a,c\},X\}$

  • $𝒫(X)$, the discrete topology (8 elements)

• Ex. Not topologies on $\{a,b,c\}$: $\{\{a\},\{c\},\{a,b\},\{a,c\}\}$ (misses $\mathrm{\varnothing },X$); $\{\mathrm{\varnothing },\{a\},\{b\},X\}$ (no union); $\{\mathrm{\varnothing },\{a,b\},\{a,c\},X\}$ (no intersection).

• Def. $𝒯$ is coarser than ${𝒯}^{\prime }$ (${𝒯}^{\prime }$ finer) if $𝒯\subset {𝒯}^{\prime }$. Always ${𝒯}_{\mathrm{triv}}\subset 𝒯\subset {𝒯}_{\mathrm{disc}}$.

  Note. A partial order, not a total one: neither Sierpiński topology refines the other.

• Def. Cofinite topology: ${𝒯}_{\mathrm{cof}}=\{U\subset X\mid X-U\text{ finite}\}\cup \{\mathrm{\varnothing }\}$. It is a topology.

  Proof. (2): if some $U_{\beta }\ne \mathrm{\varnothing }$ then $X-\bigcup U_{\alpha }\subset X-U_{\beta }$ is finite. (3): De Morgan turns $X-\bigcap U_i$ into a finite union of finite sets.

• Rmk. $X$ finite $\Rightarrow {𝒯}_{\mathrm{cof}}={𝒯}_{\mathrm{disc}}$;  $X$ infinite $\Rightarrow {𝒯}_{\mathrm{cof}}⊊{𝒯}_{\mathrm{disc}}$ (singletons are not cofinite). On $\mathbb{N}$: ${𝒯}_{\mathrm{triv}}⊊{𝒯}_{\mathrm{cof}}⊊{𝒯}_{\mathrm{disc}}$.

• Def. A metric on $X$ is $d:X\times X\to ℝ$ with

  1. $d(x,y)\ge 0$, with $d(x,y)=0\iff x=y$;

  2. $d(x,y)=d(y,x)$;

  3. $d(x,z)\le d(x,y)+d(y,z)$.

  $\epsilon$-ball: .

• $\mathrm{★}$ Def. Metric topology ${𝒯}_d$: $U$ open $\iff$ for each $x\in U$ there is $\epsilon >0$ with $B_d(x,\epsilon )\subset U$. This is a topology.

  Proof. (3) is where finiteness enters: take $\epsilon =\min \{{\epsilon }_1,\mathrm{\dots },{\epsilon }_n\}>0$. An infinite family would give $inf=0$.

1.2  (§13) Basis for a topology

• $\mathrm{★}$ Def. $ℬ$, a collection of subsets of $X$, is a basis if

  1. each $x\in X$ lies in some $B\in ℬ$;

  2. $x\in B_1\cap B_2$ $\Rightarrow$ there is $B_3\in ℬ$ with $x\in B_3\subset B_1\cap B_2$.

• Def. Topology generated by $ℬ$: $U\in 𝒯\iff \forall x\in U\exists B\in ℬ:x\in B\subset U.$ This is a topology.

• Ex. Open discs in ${ℝ}^2$;  open rectangles $(a,b)\times (c,d)$;  singletons $\{x\}$ (generating ${𝒯}_{\mathrm{disc}}$).

  Proof. For rectangles, $B_1\cap B_2$ is again a basis element or empty — condition (2) is free. For discs it needs the triangle inequality.

• Lem. Let $ℬ$ be a basis for $𝒯$. Then (1) each $B\in ℬ$ is open; (2) every open $U$ is a union of basis elements: $U={\bigcup }_{x\in U}{B}_x$.

  Note. The gluing trick — assemble local choices $B_x$ into one global object. It recurs throughout the course; name it now.

• $\mathrm{★}$ Lem. (Recognition criterion.) Let $𝒞$ be a collection of open sets in $(X,𝒯)$ such that for every open $U$ and every $x\in U$ there is $C\in 𝒞$ with $x\in C\subset U$. Then $𝒞$ is a basis for $𝒯$.

  Note. The workhorse for identifying bases. Use it for products, subspaces, metric subspaces.

• $\mathrm{★}$ Lem. (Comparison.) $ℬ,{ℬ}^{\prime }$ bases for $𝒯,{𝒯}^{\prime }$. TFAE:

  1. $𝒯\subset {𝒯}^{\prime }$;

  2. for each $B\in ℬ$ and $x\in B$ there is $B^{\prime }\in {ℬ}^{\prime }$ with $x\in B^{\prime }\subset B$.

• Caution. $𝒯\subset {𝒯}^{\prime }$ does not require $ℬ\subset {ℬ}^{\prime }$ — only that ${ℬ}^{\prime }$ has smaller sets near each point.

• Cor. $ℬ$ and ${ℬ}^{\prime }$ generate the same topology $\iff$ both conditions hold: for each $x\in B\in ℬ$ there is $B^{\prime }\in {ℬ}^{\prime }$ with $x\in B^{\prime }\subset B$, and for each $x\in B^{\prime }\in {ℬ}^{\prime }$ there is $B\in ℬ$ with $x\in B\subset B^{\prime }$.

• Ex. Discs and rectangles generate the same topology on ${ℝ}^2$, namely the metric topology.

• Def. Standard topology on $ℝ$: basis ; equals ${𝒯}_d$ for $d(x,y)=|y-x|$.

• Def. Lower limit topology ${𝒯}_{\mathrm{\ell }}$ on $ℝ$: basis ; write ${ℝ}_{\mathrm{\ell }}$.

• Lem. ${𝒯}_d⊊{𝒯}_{\mathrm{\ell }}$.

  Proof. $\subset$: given $x\in (a,b)$ take $[x,b)$. Strict: no $(c,d)$ sits inside $[a,b)$ around $x=a$, since $(c+a)/2$ escapes.

• Def. A subbasis $𝒮$ is a collection of subsets with ${\bigcup }_{S\in 𝒮}S=X$. Its associated basis is $ℬ=\{S_1\cap \mathrm{\cdots }\cap S_n\mid S_i\in 𝒮,n\ge 1\}$; the topology generated by $𝒮$ is the one generated by $ℬ$. Always $𝒮\subset ℬ\subset 𝒯$.

  Proof. $ℬ$ satisfies the basis axioms: (2) holds because $B_1\cap B_2$ is again a finite intersection of subbasis elements. The bound $n\ge 1$ avoids the empty intersection.

1.3  (§14) The order topology

• Setting. $X$ a simply ordered set (linear order ), with more than one element. Four intervals for :

  $$\underset{\text{open}}{\underbrace{(a,b)}},\underset{\text{half-open}}{\underbrace{(a,b],[a,b)}},\underset{\text{closed}}{\underbrace{[a,b]}}.$$

  Note. Nothing here is $ℝ$: these are intervals in an arbitrary ordered set. The name “open” is a promise we are about to keep.

• $\mathrm{★}$ Def. The order topology on $X$ is generated by the basis $ℬ$ consisting of

  1. all open intervals $(a,b)$;

  2. all $[a_0,b)$, if $X$ has a smallest element $a_0$;

  3. all $(a,b_0]$, if $X$ has a largest element $b_0$.

  No smallest element $\Rightarrow$ no sets of type (2); no largest $\Rightarrow$ none of type (3).

  Proof. Basis axiom (1): $a_0$ (if any) lies in every set of type (2), $b_0$ in every set of type (3), any other $x$ in some $(a,b)$ — pick . Axiom (2) is free: the intersection of two sets from the list is again such a set, or empty.

• Ex. $ℝ$ with its usual order: the order topology is the standard topology. (No largest or smallest element, so only type (1) survives.)

  Note. So the standard topology on $ℝ$ was an order topology all along — we just met it first through the basis $\{(a,b)\}$. Say this; it is the point of the section.

• Ex. ${\mathbb{Z}}_{+}$ (smallest element $1$, no largest): the order topology is ${𝒯}_{\mathrm{disc}}$. Indeed $\{n\}=(n-1,n+1)$ for $n>1$, and $\{1\}=[1,2)$ is of type (2).

• Ex. $ℝ\times ℝ$ in the dictionary order ( iff , or $x_1=x_2$ and ). No largest or smallest, so the basis is all $(a\times b,c\times d)$. The vertical segments $\{a\}\times (b,d)$ alone already form a basis.

  Note. Draw both shapes: for the interval is a “staircase” — the top of the line $x=a$, all lines strictly between, the bottom of the line $x=c$. Punchline: with the segments as basis, each vertical line is clopen and carries its usual topology, i.e. this is ${ℝ}_{\mathrm{disc}}\times ℝ$.

• $\mathrm{★}$ Ex. $X=\{1,2\}\times {\mathbb{Z}}_{+}$ in the dictionary order; write $a_n=1\times n$, $b_n=2\times n$, so

  $$a_1,a_2,a_3,\mathrm{\dots };b_1,b_2,b_3,\mathrm{\dots }$$

  The order topology is not discrete. Every $\{a_n\}$ and every $\{b_n\}$ with $n>1$ is open — but $\{b_1\}$ is not.

  Proof. $b_1$ has no immediate predecessor: any basis element containing $b_1$ is some $(a_m,d)$, and that already contains $a_{m+1}$. So no basis element about $b_1$ shrinks to $\{b_1\}$.

  Note. The example to remember: “two copies of ${\mathbb{Z}}_{+}$, one after the other”. It is where order topologies stop looking like $ℝ$.

• Def. The four rays at $a\in X$:

  $$(a,+\mathrm{\infty }),(-\mathrm{\infty },a)\text{(}\text{open}\text{)};[a,+\mathrm{\infty }),(-\mathrm{\infty },a]\text{(}\text{closed}\text{)}.$$

• Lem. Open rays are open.

  Proof. For $(a,+\mathrm{\infty })$: if $X$ has a largest element $b_0$ it equals the basis element $(a,b_0]$; if not, it is ${\bigcup }_{x>a}(a,x)$. Dually for $(-\mathrm{\infty },a)$.

• $\mathrm{★}$ Thm. The open rays form a subbasis for the order topology.

  Proof. $\subset$: they are open, so they generate no more. $\supset$: every basis element is a finite intersection of them — $(a,b)=(-\mathrm{\infty },b)\cap (a,+\mathrm{\infty })$, while $[a_0,b)=(-\mathrm{\infty },b)$ and $(a,b_0]=(a,+\mathrm{\infty })$ are themselves rays.

  Note. Same move as for the product topology: a small subbasis replaces a bulky basis. Two rays cut out an interval exactly as two slabs cut out a rectangle.

• Caution. For $Y\subset X$ the subspace topology and the order topology of $Y$ need not agree. E.g. $Y=[0,1)\cup \{2\}\subset ℝ$: $\{2\}$ is open in the subspace topology, but in the order topology on $Y$ every basis element about $2$ is some $(a,2]$, which meets $[0,1)$.

  Note. Defer the discussion to §16; but plant the warning here, while the definition is on the board.