Topology Review

Everything from the section, arranged for lookup: the definitions, the named theorems, the standard spaces, the counterexamples, and the invariants with what each one can prove.

A metric satisfies non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Open balls give open sets, and different metrics can produce the same topology - $d_1$, $d_2$ and $d_\infty$ on $\mathbb{R}^n$ all do, which is the motivation for dropping the metric entirely.

A topology is a family of subsets containing $\varnothing$ and $X$, closed under arbitrary unions and finite intersections. The asymmetry is forced: $\bigcap(-\frac1n,\frac1n) = \{0\}$ is not open.

Closed sets dualize: arbitrary intersections and finite unions. The dual counterexample is $\bigcup[\frac1n,1] = (0,1]$.

Open and closed are not opposites. All four combinations occur in $\mathbb{R}$: $(0,1)$, $[0,1]$, $\varnothing$, $[0,1)$.

Topology	Open sets
Discrete	all subsets; finest
Trivial	only $\varnothing$ and $X$; coarsest
Standard on $\mathbb{R}$	unions of open intervals
Cofinite	$\varnothing$ plus sets with finite complement
Lower limit	generated by $[a,b)$; strictly finer than standard

Counts of distinct topologies: 1, 4, 29, 355, 6942 on 1 to 5 points. No formula is known.

A basis is a family whose unions give every open set. To generate a topology it must cover $X$ and satisfy: for $x\in B_1\cap B_2$ there is $B_3$ with $x\in B_3\subseteq B_1\cap B_2$.

The subspace topology on $A$ is $\{U\cap A\}$. So openness is always relative: $[0,\frac12)$ is open in $[0,1]$ and not in $\mathbb{R}$, and $\mathbb{Z}\subseteq\mathbb{R}$ is discrete.

Hereditary: Hausdorff, metrizable, second countable. Not hereditary: compactness, connectedness. Closed subspaces are the well-behaved case.

$$f \text{ continuous} \iff f^{-1}(V) \text{ open for every open } V$$

Equivalent forms: preimages of closed sets are closed; preimages of basis elements are open (cheapest); $f(\bar A)\subseteq\overline{f(A)}$.

Continuity constrains preimages only. $x^2$ sends the open $(-1,1)$ to $[0,1)$.

A homeomorphism is a continuous bijection with continuous inverse. The inverse condition is not redundant: $[0,2\pi)\to S^1$ is a continuous bijection that is not a homeomorphism. It comes free when the domain is compact and the target Hausdorff.

To prove $X\not\cong Y$, exhibit a differing invariant. Matching invariants prove nothing.

Every point is interior, boundary, or exterior relative to $A$.

Always: $\overline{A\cup B}=\bar A\cup\bar B$ and $\operatorname{int}(A\cap B)=\operatorname{int}A\cap\operatorname{int}B$.

Containment only: $\overline{A\cap B}\subseteq\bar A\cap\bar B$ and $\operatorname{int}(A\cup B)\supseteq\operatorname{int}A\cup\operatorname{int}B$.

Dense means $\bar A=X$; nowhere dense means $\operatorname{int}\bar A=\varnothing$. $\mathbb{Q}$ is dense in $\mathbb{R}$ with $\partial\mathbb{Q}=\mathbb{R}$; $\mathbb{Z}$ and the Cantor set are nowhere dense.

Connected means no separation into two nonempty disjoint open sets. Equivalently only $\varnothing$ and $X$ are clopen, or every continuous map to a discrete space is constant.

The connected subsets of $\mathbb{R}$ are exactly the intervals.

Continuous images of connected sets are connected, which gives the intermediate value theorem in one line. $\mathbb{Q}$ is totally disconnected.

Path-connected implies connected, not conversely. The topologist’s sine curve has 1 component and 2 path components. The two notions agree when the space is locally path-connected, which covers open subsets of $\mathbb{R}^n$ and all manifolds.

$$\text{convex}\implies\text{star-shaped}\implies\text{path-connected}\implies\text{connected}$$

all strict.

Compact means every open cover has a finite subcover. It is the topological stand-in for finiteness.

Heine–Borel. In $\mathbb{R}^n$ only: compact $\iff$ closed and bounded.

In a general metric space the correct statement is complete + totally bounded, and $\{e_n\}\subseteq\ell^2$ is bounded without being totally bounded.

Consequences: continuous images are compact; the extreme value theorem; closed subspaces of compact spaces are compact; compact subsets of Hausdorff spaces are closed; continuous functions on compact metric spaces are uniformly continuous; products of compact spaces are compact (Tychonoff).

Compactness is not hereditary: $(0,1)$ inside $[0,1]$.

In metric spaces, compact = sequentially compact = complete and totally bounded = limit point compact. None of these equivalences holds in general.

Product topology. Basis of boxes $U\times V$; the coarsest making the projections continuous. A map into a product is continuous iff its components are.

Quotient topology. $V$ open iff $q^{-1}(V)$ is open; the finest making $q$ continuous. A map out is continuous iff its composite with $q$ is. Quotients can destroy Hausdorff.

Gluing a square:

Gluing	Result	Orientable	Boundary circles
one pair, straight	cylinder	yes	2
one pair, flipped	Möbius strip	no	1
both straight	torus	yes	0
one flipped	Klein bottle	no	0
both flipped	$\mathbb{RP}^2$	no	0

$$T_4\implies T_3\implies T_2\implies T_1\implies T_0$$

$T_0$: distinguishable. $T_1$: singletons closed. $T_2$ Hausdorff: disjoint neighbourhoods, hence unique limits. $T_3$: point from closed set. $T_4$ normal: two closed sets, which is what Urysohn’s lemma needs.

Every metric space is $T_4$, so failing any axiom rules out metrizability. The cofinite topology on an infinite set is $T_1$ and not $T_2$. Urysohn metrization: regular Hausdorff plus a countable basis implies metrizable. Compact + Hausdorff implies normal.

A closed surface is a compact 2-manifold without boundary.

Classification. Every closed connected surface is the sphere, a connected sum of $g$ tori, or a connected sum of $k$ projective planes - determined by orientability and genus.

$$\chi = 2-2g \ \text{(orientable)}, \qquad \chi = 2-k \ \text{(non-orientable)}$$

Connected sum: $\chi(S_1\# S_2)=\chi(S_1)+\chi(S_2)-2$, the sphere is the identity, $\mathbb{RP}^2\#\mathbb{RP}^2$ is the Klein bottle, and once non-orientable, adding a torus equals adding two crosscaps.

$\chi$ alone is not enough: the torus and Klein bottle both have $\chi=0$.

Edge words: sphere $aa^{-1}$, torus $aba^{-1}b^{-1}$, $\mathbb{RP}^2$ $aa$, Klein bottle $abab^{-1}$.

$\chi=V-E+F$, invariant because refining adds a vertex and an edge, or an edge and a face. It is a homotopy invariant, so a disc and a point share $\chi=1$.

Applications: exactly five Platonic solids from $\frac1p+\frac1q>\frac12$; planar graphs satisfy $E\le3V-6$, ruling out $K_5$; Poincaré–Hopf gives the hairy ball theorem since $\chi(S^2)=2\ne0$.

A homotopy is a continuous $H:X\times[0,1]\to Y$ joining two maps. Homotopy equivalence needs only the composites to be homotopic to the identities, making it coarser than homeomorphism: it forgets dimension, compactness and orientability.

Deformation retracts are the practical tool: $\mathbb{R}^2\setminus\{0\}$, the annulus, the solid torus and the Möbius strip all retract to a circle.

$\pi_1(X,x_0)$ is the group of path-homotopy classes of loops under concatenation, with associativity holding only up to homotopy.

Space	$\pi_1$
$\mathbb{R}^n$, discs, cones	trivial
$S^1$, annulus, punctured plane, solid torus, Möbius strip	$\mathbb{Z}$
$S^n$, $n\ge2$	trivial
Torus	$\mathbb{Z}\times\mathbb{Z}$
Figure eight	free on 2 generators, non-abelian
$\mathbb{RP}^2$	$\mathbb{Z}/2$

Simply connected means path-connected with trivial $\pi_1$ - the hypothesis in Cauchy’s theorem. $S^2$ is simply connected and not contractible.

$\pi_1$ is functorial, multiplicative on products, and computable by van Kampen. It separates the torus from the Klein bottle where $\chi$ cannot, and it proves no retraction $D^2\to S^1$ exists.

Brouwer. Every continuous self-map of a closed ball has a fixed point.

$n=1$ is the intermediate value theorem applied to $f(x)-x$. $n=2$ follows from the non-existence of a retraction $D^2\to S^1$, hence from $\pi_1(S^1)=\mathbb{Z}$. Fails on the annulus (a hole), on the open disc (not compact), and for discontinuous maps.

Banach. A contraction on a complete metric space has a unique fixed point, reachable by iteration.

Brouwer is topological, existence only, non-constructive. Banach is metric, unique, and gives an algorithm.

Consequences: hairy ball, Borsuk–Ulam, ham sandwich, Nash equilibria, Perron–Frobenius.

Object	Shows
$\bigcap(-\frac1n,\frac1n)=\{0\}$	intersections must be finite
$\bigcup[\frac1n,1]=(0,1]$	unions of closed sets need not be closed
$[0,1)$	neither open nor closed
$\mathbb{Q}$ in $\mathbb{R}$	$\partial A$ can be the whole space; totally disconnected
Cofinite on an infinite set	$T_1$ without $T_2$; not metrizable
Sierpiński space	$T_0$ without $T_1$
Trivial topology	limits not unique; compact sets not closed
$[0,2\pi)\to S^1$	continuous bijection, not a homeomorphism
$(0,1)\cong\mathbb{R}$	boundedness is not topological
$(0,1)$ in $[0,1]$	compactness is not hereditary
$\{e_n\}\subseteq\ell^2$	bounded but not totally bounded
Topologist’s sine curve	connected, not path-connected
Cantor set	compact, uncountable, totally disconnected, nowhere dense
Möbius strip vs cylinder	one reversed gluing changes orientability
Torus vs Klein bottle	same $\chi$, different spaces
$S^2$	simply connected, not contractible
Annulus rotation	Brouwer needs the right shape

• Thinking open and closed are opposites. $[0,1)$ is neither; $\varnothing$ and $X$ are both.
• Forgetting “open in what?” $[0,\frac12)$ is open in $[0,1]$ and not in $\mathbb{R}$.
• Requiring arbitrary intersections to be open, or arbitrary unions of closed sets to be closed.
• Defining continuity with images instead of preimages.
• Omitting continuity of the inverse in a homeomorphism.
• Treating closed and bounded as compact outside $\mathbb{R}^n$.
• Assuming connected implies path-connected, without local path-connectedness.
• Assuming compactness or connectedness is hereditary.
• Concluding two spaces are homeomorphic because invariants agree.
• Using $\chi$ alone to classify a surface, ignoring orientability.
• Confusing simply connected with contractible. $S^2$ is the first and not the second.
• Expecting Brouwer to give uniqueness, or Banach to work without a contraction.
• Confusing the solid torus with the torus surface; their $\pi_1$ are $\mathbb{Z}$ and $\mathbb{Z}\times\mathbb{Z}$.