A Stone space is a compact Hausdorff space that is totally disconnected, equivalently a compact Hausdorff space with a basis of clopen sets. The topology on a type space gives an important example.

A non-Hausdorff manifold is a locally Euclidean topological space for which distinct points need not have disjoint neighborhoods. The real line with two origins is a basic example. Nonseparated algebraic schemes exhibit analogous gluing phenomena, but are not generally topological manifolds.

Orientability asks whether local choices of handedness can agree everywhere on a manifold. On a smooth manifold, an orientation of a smooth manifold chooses an orientation of a vector space for each tangent space, continuously with position. An ordered basis describes the same handedness as another one when the change-of-basis determinant is positive. An oriented atlas implements this condition through positive Jacobian determinants of the smooth transition maps.

A paper strip makes the obstruction visible. Join its short ends without twisting: the resulting untwisted band admits a consistent choice of unit normal. Add a half-turn before joining: on the Möbius band, following the core circle of a Möbius band reverses that choice when you first return to the same point. A local decision about handedness has encountered a global contradiction. The normal reversal around the Möbius core makes this precise for an explicit model.

Both rectangles have coordinates $(u,v)\in [0,1]\times [-1,1]$. The untwisted band identifies $(0,v)$ with $(1,v)$; the Möbius band identifies $(0,v)$ with $(1,-v)$. Across a seam the corresponding local coordinate changes preserve or reverse $v$, giving determinants $+1$ and $-1$. This sign explains the handedness change; the drawing also tracks the two boundary circles of the untwisted band and the single boundary circle of the Möbius band.

For a smooth topological surface embedded in ${\mathbb{R}}^3$, choosing an orientation of a surface is equivalent to choosing a continuous unit normal. This picture is specific to surfaces in three-dimensional space: the general definition uses compatible local orientations, without requiring an embedding. Orientability is a topological invariant, so a homeomorphism cannot turn a Möbius band into a untwisted band. The sphere and torus are orientable surfaces; the Klein bottle is not. ^[ref]

The distinction matters when a sign has geometric meaning. A signed flux surface integral needs an orientation of a surface, and Stokes theorem relates that choice to the outward-normal-first boundary orientation. Ordinary unsigned surface area does not require orientability. Thus a Möbius band still has a perfectly meaningful area even though it has no continuous global unit normal.

A connected orientable manifold of positive dimension has two possible orientations: reversing every local choice gives the other one. If a manifold has several connected components, each component can be oriented independently when it is orientable. For topological surfaces, a useful obstruction criterion is the presence of a Möbius band: a surface is nonorientable exactly when it contains such a band. ^[ref]

An end of a noncompact topological space describes a coherent way to escape every compact subset. For a cofinal compact exhaustion, when one exists, $K_1\subseteq K_2\subseteq \cdots$, it is represented by nested connected components of the complements. A connected noncompact manifold is disconnected at infinity when it has more than one end.

A topological invariant is a property or mathematical object preserved by homeomorphisms of topological spaces. Connectedness and compactness are examples. The fundamental group is an invariant up to group isomorphism; it is also preserved up to isomorphism by homotopy equivalence, a stronger invariance property which does not hold for every topological invariant.

A recurrent point of a discrete dynamical system returns arbitrarily close to its starting state at arbitrarily late positive times. In a metric space, this means that there is a strictly increasing sequence $n_j\to \infty$ withEquivalently, every neighborhood of $x$ contains $T^{n}x$ for arbitrarily large $n$. A finite early return near the point does not suffice.

Every periodic point is recurrent: if $T^{p}x=x$, take $n_j=jp$. An irrational rotation of the circle gives recurrent points that are not periodic. For $T(\theta )=\theta +\alpha (\mathrm{mod}1)$ with irrational $\alpha$, the pigeonhole principle applied to $0,\alpha ,\dots ,N\alpha$ modulo one supplies an integer $1\le q\le N$ with distance from $q\alpha$ to an integer at most $1/N$. These approximating $q$ must be unbounded, since no fixed positive multiple of an irrational number is an integer. An increasing subsequence therefore gives the required returns for every starting angle.

Recurrence concerns returning close, whereas convergence to an attracting equilibrium point need not give recurrence of any other starting state. The map $T(x)=x/2$ has zero as a recurrent point, but a nonzero $x$ converges to zero rather than back to itself.

A subset $S\subseteq \mathbb{Z}$ is syndetic if it has bounded gaps: some positive integer $L$ makes every block $\{n,n+1,\dots ,n+L-1\}$ meet $S$. The quantifier applies at every integer location, including negative ones. A corresponding one-sided definition uses $S\subseteq \mathbb{N}$ and nonnegative starting locations.

For example, the even integers are syndetic with $L=2$, while the positive perfect squares are not: the gap between $m^2$ and $(m+1{)}^2$ is $2m+1$ and grows without bound. In a one-sided syndetic set, partitioning the first $N$ integers into blocks of length $L$ gives at least $\lfloor N/L\rfloor$ elements, so its lower asymptotic density is at least $1/L$.

Positive lower density alone does not ensure bounded gaps. Remove from $\mathbb{N}$ each block $\{2^k,\dots ,2^k+k-1\}$ for $k\ge 1$. The removed blocks have unbounded lengths, but at most ${\sum }_{k\le {\log }_{2}N}k=O((\log N{)}^2)$ removed points lie below $N$. The remaining set has asymptotic density one and is not syndetic. Thus a condition on every interval is stronger than a condition on the long-run proportion.

A continuous map on a nonempty topological space is point-transitive if it has a dense forward orbit. This is one convention for topological transitivity, but on spaces with isolated points it must be distinguished from the open-set intersection definition. On a nonempty compact metric space, the positive-time open-set intersection property implies point transitivity: for each member $U_j$ of a countable basis of a topology, ${\bigcup }_{n\ge 1}{f}^{-n}(U_j)$ is open and dense, and the Baire category theorem supplies a point in their intersection. Conversely, a surjective point-transitive continuous map has every tail of its dense orbit dense, since $f^k(X)=X$ and continuity sends its orbit closure into the closure of that tail. The map $f(1/j)=1/(j+1)$, $f(0)=0$ on $\{0\}\cup \{1/j:j\ge 1\}$ is point-transitive but fails the open-set intersection property.

Two continuous maps are topologically conjugate if a homeomorphism $H$ between their state spaces satisfies the displayed intertwining identity. It is conjugate maps with a topology-preserving conjugating map. On compact metric spaces, pulling back a compatible metric through $H$ preserves every Bowen metric, and hence topological entropy.

A family $\mathcal{F}$ of maps between metric spaces is equicontinuous when, near each point, one input tolerance controls the output variation uniformly for every $f\in \mathcal{F}$. Uniform equicontinuity uses one input tolerance over the whole domain.

A family $\mathcal{F}$ of maps between metric spaces is equicontinuous at $x$ when, for every $\epsilon >0$, one $\delta >0$ works for all $f\in \mathcal{F}$: $d(x,y)<\delta$ implies $d(f(x),f(y))<\epsilon$.

In a positive-time patch, proper-time first integrals give $\dot{x}=p{t}^2$ and ${\dot{t}}^2=t^2+p^2{t}^4$. If $p\ne 0$, the unparametrized timelike geodesics are $t^2=(x-x_c{)}^2-p^{-2}$ on an appropriate branch; in the expanded quadratic form their discriminant is positive. If $p=0$, there is the separate family $x=x_c$, $t=t_{\ast }{e}^{\pm \tau }$. A description using only the nonvertical quadratic family omits genuine timelike geodesics.

For $d{s}^2=(1+f(x){)}^{2}d{x}^2/(1+x^2)-(1+x^2)d{t}^2$, with smooth odd $f$ and $|f|<1$, the conserved energy gives $dt/d\tau =E/(1+x^2)$ and $(dx/d\tau {)}^2=(E^2-1-x^2)/(1+f{)}^2$. Both half-cycle integrals over $[-\sqrt{E^2-1},\sqrt{E^2-1}]$ contain an even weight times $1+f$. The odd contribution vanishes, leaving the unperturbed half-period $\pi$ in each clock. The waveforms need not be sinusoidal, but both periods are unchanged for $E>1$.

The bispectrum is the Fourier transform of a third-order cumulant or correlation function, with the convention specified for the process or field. It measures third-order dependence among frequency triples; the primordial bispectrum is a cosmological random-field application.

A second-order stationary time series decomposes into a deterministic component and a one-sided square-summable linear filter of a stationary time series of its mutually orthogonal linear innovations. Normalize $c_0=1$. For a causal invertible ARMA process with no deterministic component, the driving white noise is this innovation process: observations and innovations span the same closed past subspace, while the next innovation is orthogonal to it.

With a fixed gain $0<K<1$, exponential smoothing recursively combines the newest observation with the preceding estimate. Iteration assigns geometric weights $K(1-K{)}^j$ to older observations, together with a diminishing weight on the initial estimate. The steady-state local-level Kalman gain gives a probabilistic interpretation of this recursion.

For independent identically distributed continuous observations, count strict local maxima and minima at interior positions. Each turning indicator has probability $2/3$, adjacent indicators have joint probability $5/12$, indicators two positions apart have joint probability $9/20$, and more distant indicators are independent. Summing these covariances gives the displayed variance for $T\ge 4$. The indicators form a bounded 2-dependent sequence, giving an asymptotically normal standardized count. Too few turns suggests persistent trends; too many suggests alternation. Ties and dependent observations require a different calibration.

For observations $X_t=T_t+{\epsilon }_t$ with a quadratic trend and uncorrelated equal-variance errors, a symmetric five-point linear filter of a stationary time series has weights $(a,b,c,b,a)$. Reproducing every quadratic requires $2a+2b+c=1$ and $8a+2b=0$. Substitution gives $b=-4a$, $c=1+6a$ and error variance ${\sigma }^2(70a^2+12a+1)$. Its unique minimum occurs at $a=-3/35$, producing the displayed weights and variance $17{\sigma }^2/35$. Negative weights are allowed: this is a linear weighted smoother, not a convex average. Applying a linear filter to a deterministic trend plus stationary errors does not assert that the complete observation process is stationary.

A symmetric three-point smoother with weights $(a,b,a)$ reproduces constants only if $2a+b=1$. Reproducing quadratic trends also requires $2a=0$, so $a=0$ and $b=1$. Thus the identity estimator $X_t$ is unbiased, but there is no nontrivial three-point linear smoother reproducing every quadratic. An assertion excluding even the identity estimator is false. This clarifies why the five-point symmetric quadratic trend smoother is the first symmetric nontrivial example.

A lag of a time series is a separation between observation indices. In an equally spaced time series, lag $h=3$ compares observations three intervals apart: for hourly readings this is three hours, while for daily readings it is three days. Positive and negative lags describe the same separation in opposite directions. The autocovariance and autocorrelation of a real weakly stationary process are even functions of lag.

Coordinating concurrent activities so that required ordering and visibility rules hold. Mutual exclusion is one form of synchronization; waiting for another software thread to finish is another.