0.1  (§18) Continuous functions

• $\mathrm{★}$ Def. $f:X\to Y$ is continuous if $f^{-1}(V)$ is open in $X$ for every open $V\subset Y$. A continuous function is called a map.

  Note. Refer back to §2: preimages respect $\cup$, $\cap$, $-$. Images do not. That is the entire reason the definition takes this form.

• Lem. Composites of maps are maps.

  Proof. ${(g\circ f)}^{-1}(W)=f^{-1}(g^{-1}(W))$.

• $\mathrm{★}$ Lem. It is enough to test a basis: $f$ continuous $\iff f^{-1}(B)$ open for all $B\in ℬ$. And enough to test a subbasis: $\iff f^{-1}(S)$ open for all $S\in 𝒮$.

  Proof. Because $f^{-1}$ commutes with unions (basis) and with finite intersections (subbasis).

• Ex. Metric spaces: continuity $\iff$ for all $x$ and $\epsilon >0$ there is $\delta >0$ with . The $\epsilon$–$\delta$ definition, recovered.

• $\mathrm{★}$ Ex. $\mathrm{id}:ℝ\to {ℝ}_{\mathrm{\ell }}$ is not continuous ($[a,b)$ is not open in $ℝ$); $\mathrm{id}:{ℝ}_{\mathrm{\ell }}\to ℝ$ is continuous.

• Def. $f$ is continuous at $x$ if for each neighborhood $V$ of $f(x)$ there is a neighborhood $U$ of $x$ with $f(U)\subset V$.

• Thm. $f$ continuous $\iff$ $f$ continuous at every $x\in X$.

  Proof. $\Leftarrow$ is the gluing trick again: $f^{-1}(V)={\bigcup }_{x\in f^{-1}(V)}{U}_x$.

• $\mathrm{★}$ Thm. TFAE:

  1. $f$ is continuous;

  2. $f(\overline{A})\subset \overline{f(A)}$ for every $A\subset X$;

  3. $f^{-1}(B)$ is closed for every closed $B\subset Y$.

  Proof. $(1)\Rightarrow (2)$ via the neighborhood criterion for closure; $(2)\Rightarrow (3)$ apply to $A=f^{-1}(B)$; $(3)\Rightarrow (1)$ complement.

• $\mathrm{★}$ Def. $f:X\to Y$ is a homeomorphism if it is bijective and both $f$ and $f^{-1}$ are continuous. Then $X\cong Y$, topologically equivalent.

• Lem. $\cong$ is an equivalence relation (identity, inverse, composite).

• Lem. For $f$ bijective, TFAE: $f$ homeomorphism; $U$ open in $X\iff f(U)$ open in $Y$; $V$ open in $Y\iff f^{-1}(V)$ open in $X$.

  Note. So a homeomorphism is a bijection ${𝒯}_X\to {𝒯}_Y$ of the topologies, not just of the points.

• Def. A topological property is one expressible in terms of points and open sets; such properties are preserved by homeomorphism. E.g. finiteness, discreteness, Hausdorff, and later compactness and connectedness.

• Ex. $(0,1)\cong (a,b)$ via $f(x)=(1-x)a+xb$.

• Ex. $(-1,1)\cong ℝ$ via $f(x)=x/(1-x^2)$,   $f^{-1}(y)=2y/\left(1+\sqrt{1+4y^2}\right)$. Hence every open interval $\cong ℝ$.

• $\mathrm{★}$ Caution. Continuous bijection $\ne$ homeomorphism.

  • $\mathrm{id}:{ℝ}_{\mathrm{\ell }}\to ℝ$.

  • $f:[0,1)\to S^1$, $f(t)=(\cos 2\pi t,\sin 2\pi t)$. Here $U=[0,\frac{1}{2})$ is open in $[0,1)$ but $f(U)$ is not open in $S^1$: no neighborhood of $(1,0)$ lies inside it.

Constructing maps.

• Thm. The subspace topology on $A\subset X$ is the coarsest making the inclusion $i:A\to X$ continuous.

• Cor. Restrictions $f|A$ of maps are maps. Lem. Corestrictions $X\to B$ (with $f(X)\subset B$) are maps.

• Def. $f:X\to Y$ is an embedding if the corestriction $X\to f(X)$ is a homeomorphism onto the subspace $f(X)$.

• Lem. $f$ is an embedding $\iff$ $f=j\circ h$ with $h$ a homeomorphism onto a subspace and $j$ the inclusion. Embeddings are injective.

• Ex. $[0,1)\to {ℝ}^2$, $t\mapsto (\cos 2\pi t,\sin 2\pi t)$: injective and continuous, not an embedding.

• Lem. Maps to a one-point space are continuous; hence constant maps are continuous.

• $\mathrm{★}$ Thm. (Pasting lemma.)

  • $X={\bigcup }_{\alpha \in J}{U}_{\alpha }$ with all $U_{\alpha }$ open: $f$ continuous $\iff$ every $f|U_{\alpha }$ continuous.

  • $X=A_1\cup \mathrm{\cdots }\cup A_n$ with all $A_i$ closed (finitely many): $f$ continuous $\iff$ every $f|A_i$ continuous.

  Proof. Open case: $f^{-1}(V)=\bigcup {(f|U_{\alpha })}^{-1}(V)$, a union of open sets. Closed case: same with closed sets — and now finiteness is essential.

• $\mathrm{★}$ Thm. (Maps into a product.) $f:A\to X\times Y$ is continuous $\iff$ both components $f_1={\pi }_{1}f$ and $f_2={\pi }_{2}f$ are continuous.

  Proof. Subbasis criterion: $f^{-1}(U\times Y)=f_1^{-1}(U)$, $f^{-1}(X\times V)=f_2^{-1}(V)$.

• Cor. The product topology is the coarsest topology making both projections continuous.

0.2  (§19) The product topology (general)

• $\mathrm{★}$ Def. On ${\prod }_{\alpha \in J}{X}_{\alpha }$ the product topology is generated by the subbasis $𝒮=\{{\pi }_{\beta }^{-1}(U_{\beta })\mid \beta \in J,U_{\beta }\subset X_{\beta }\text{ open}\}.$

• $\mathrm{★}$ Lem. Concretely:

  • subbasis elements are ${\prod }_{\alpha }{U}_{\alpha }$ with $U_{\alpha }\ne X_{\alpha }$ for at most one $\alpha$;

  • basis elements are ${\prod }_{\alpha }{U}_{\alpha }$ with $U_{\alpha }\ne X_{\alpha }$ for finitely many $\alpha$.

  Note. Emphasise “all but finitely many coordinates unrestricted”. This is the point where students expect the naive definition and get a different one.

• Thm. $f:A\to \prod X_{\alpha }$ continuous $\iff$ every component $f_{\beta }={\pi }_{\beta }f$ is continuous.

• Cor. The product topology is the coarsest making all ${\pi }_{\beta }$ continuous.

• Thm. $A_{\alpha }\subset X_{\alpha }$ subspaces: on $\prod A_{\alpha }$, product topology $=$ subspace topology from $\prod X_{\alpha }$.

• Thm. All $X_{\alpha }$ Hausdorff $\Rightarrow \prod X_{\alpha }$ Hausdorff.

• Thm. $\overline{{\prod }_{\alpha }{A}_{\alpha }}={\prod }_{\alpha }\overline{A_{\alpha }}$.