Topological spaces

Topology keeps exactly the structure needed for continuity: open sets.

1. Topological spaces (Def. 1.1) and bases (Def. 1.2, Prop. 1.3), with standard examples (Example 1.4).
2. Closed sets, closure and interior (Def. 1.5, Prop. 1.6).
3. Continuity (Def. 1.7, Prop. 1.8) and homeomorphism (Def. 1.9).
4. New spaces: subspaces (Def. 1.10), products (Def. 1.11, Prop. 1.12) and quotients (Def. 1.13, Example 1.14).
5. Topological invariants (Check 1.15).

A topology on a set $X$ is a collection $\mathcal{T}$ of subsets (the open sets) such that $\emptyset, X\in\mathcal{T}$, arbitrary unions of open sets are open, and finite intersections of open sets are open. $(X, \mathcal{T})$ is a topological space. If $\mathcal{T}_1\subseteq\mathcal{T}_2$, $\mathcal{T}_1$ is coarser and $\mathcal{T}_2$ finer.

A collection $\mathcal{B}$ of subsets of $X$ is a basis for a topology if (i) every point lies in some $B\in\mathcal{B}$ and (ii) whenever $x\in B_1\cap B_2$ there is $B_3\in\mathcal{B}$ with $x\in B_3\subseteq B_1\cap B_2$. The topology generated by $\mathcal{B}$ consists of all unions of members of $\mathcal{B}$.

If $\mathcal{B}$ is a basis, the unions of members of $\mathcal{B}$ form a topology, and $U$ is open in it iff for every $x\in U$ there is $B\in\mathcal{B}$ with $x\in B\subseteq U$.

• Metric topology: open sets of a metric space (basis: open balls).
• Discrete (all subsets) and indiscrete ($\{\emptyset, X\}$) topologies.
• Cofinite topology: $\emptyset$ and complements of finite sets.
• Lower limit topology on $\mathbb{R}$: basis $[a, b)$; finer than the usual topology. Different metrics can give the same topology (e.g. $d_1, d_2, d_\infty$ on $\mathbb{R}^n$).

$F\subseteq X$ is closed if $X\setminus F$ is open. The closure $\overline A$ is the intersection of all closed sets containing $A$; the interior $A^\circ$ is the union of all open sets inside $A$. A point $x$ is a limit point of $A$ if every open set containing $x$ meets $A\setminus\{x\}$. $A$ is dense if $\overline A = X$.

$x\in\overline A$ iff every open set (equivalently, every basis set) containing $x$ meets $A$. Hence $\overline A = A\cup A'$ where $A'$ is the set of limit points, and $A$ is closed iff it contains its limit points.

$f: X\to Y$ is continuous if $f^{-1}(V)$ is open in $X$ for every open $V\subseteq Y$. For metric spaces this agrees with the $\varepsilon$–$\delta$ definition.

For $f: X\to Y$ the following are equivalent: (a) $f$ is continuous; (b) preimages of closed sets are closed; (c) $f(\overline A)\subseteq\overline{f(A)}$ for all $A\subseteq X$; (d) preimages of basis sets of $Y$ are open. Compositions of continuous maps are continuous.

A homeomorphism is a bijection $f: X\to Y$ with $f$ and $f^{-1}$ continuous; equivalently a bijection carrying open sets to open sets. Then $X\cong Y$. A property preserved by homeomorphisms is a topological invariant.

For $A\subseteq X$, the subspace topology is $\{U\cap A : U\text{ open in }X\}$. It is the coarsest topology making the inclusion $A\hookrightarrow X$ continuous, and $f: Z\to A$ is continuous iff it is continuous as a map into $X$.

The product topology on $X\times Y$ has basis the "open rectangles" $U\times V$ with $U, V$ open. The projections $p_X, p_Y$ are continuous. For $\mathbb{R}\times\mathbb{R}$ it is the usual topology of $\mathbb{R}^2$.

A map $f = (f_1, f_2): Z\to X\times Y$ is continuous iff $f_1 = p_X\circ f$ and $f_2 = p_Y\circ f$ are both continuous.

Given an equivalence relation $\sim$ on $X$ with quotient map $q: X\to X/{\sim}$, the quotient topology declares $V\subseteq X/{\sim}$ open iff $q^{-1}(V)$ is open in $X$. A map $g: X/{\sim}\to Y$ is continuous iff $g\circ q$ is continuous.

• $[0, 1]/(0\sim1)\cong S^1$, via $t\mapsto e^{2\pi it}$.
• The square $[0, 1]^2$ with $(x, 0)\sim(x, 1)$ and $(0, y)\sim(1, y)$ is the torus $S^1\times S^1$.
• Identifying $(0, y)\sim(1, 1 - y)$ only gives the Möbius strip; with both sides twisted, the Klein bottle.
• $D^2/\partial D^2\cong S^2$ (collapse the boundary circle to a point). Homeomorphisms out of quotients are usually proved with the compact-to-Hausdorff criterion (see Section 3).

• To show $X\cong Y$: write down a bijection and check both directions are continuous (or see Section 3: compact → Hausdorff).
• To show $X\not\cong Y$: find a topological invariant they differ in — compactness, connectedness, number of components after removing a point, Hausdorffness, countability, the fundamental group.
• Cardinality and metric quantities (lengths, boundedness) are not topological invariants.