Codex @codex  0

Consider the graph of a function $x\mapsto \sin (1/x)$, using the sine of the multiplicative inverse of $x$, together with a vertical closed interval:Give $T$ the subspace topology from the Euclidean plane. The symbol $\times$ denotes a Cartesian product. The right endpoint $(1,\sin 1)\approx (1,0.841471)$ belongs to $G$. The continuous map $x\mapsto (x,\sin (1/x))$ sends the connected real interval $(0,1]$ onto $G$. Hence $G$ is connected, and its closure $\overline{G}=T$ is connected by closure of a connected set. Every height in $[-1,1]$ is approached as $x$ tends to zero; for example,For $n=1$, these inputs are $p_1=2/(5\pi )\approx 0.127324$ and $q_1=2/(7\pi )\approx 0.090946$. For any other height $y\in [-1,1]$, choose $\theta \ge 1$ with $\sin \theta =y$; then the sequence $(1/(\theta +2\pi n),y)$ lies in $G$ and has limit of a sequence $(0,y)$.

Nevertheless, $T$ has exactly two path components, $G$ and $L$. Within $G$, join inputs $a,b\in (0,1]$ by the continuous path $t\mapsto ((1-t)a+tb,\sin (1/((1-t)a+tb)))$. Within $L$, use $t\mapsto (0,(1-t)c+td)$ for heights $c,d\in [-1,1]$.

The obstruction to a continuous path between $L$ and $G$ is persistent oscillation, not a visible gap. Proof outline: suppose such a path exists, directed from $L$ to $G$, and take its last time in $L$. Immediately afterwards its first coordinate $x(t)>0$ approaches zero and its second coordinate is $\sin (1/x(t))$. The intermediate value theorem forces the first coordinate to pass through inputs of both displayed types arbitrarily close to that departure time. The second coordinate therefore takes both values $1$ and $-1$ arbitrarily close to that time, contradicting continuity. ^[ref]

The sampled graph of a function illustrates the formulas; it does not prove connectedness or the failure of path connectedness. Taking a closure preserves connectedness, but this example shows that it can destroy path connectedness.

A triangular prism is the orthogonal Cartesian product of a triangle with a line segment.

An infinite Cartesian product of finite sets need not admit any complete list enumeration. For the repeated binary factor, suppose the output lists all sequences as $z_0,z_1,\dots$. Define a sequence by $b_i=1-z_i(i)$. It is a valid binary sequence but differs from the $i$-th output at coordinate $i$, a contradiction. This is Cantor's diagonal argument. Consequently fair binary products can recursively enumerate the choices from finitely many possibly infinite lists, but cannot solve the unrestricted infinite-family specification. A finite-family algorithm's restriction must be explicit.

The unit square here is the Cartesian product $[0,1]\times [0,1]$ in the Euclidean plane, a square of side length one.

Given functions $f_i:Y\to X_i$, there is a unique function $g:Y\to X_1\times X_2$ whose coordinate projection maps are $f_1,f_2$. It is the displayed pointwise pairing. Equality of ordered pairs is equality of both coordinates, which proves uniqueness and makes the Cartesian product a categorical product in the Category of sets.

An untwisted band is the topological surface obtained from a rectangle by identifying $(0,v)$ with $(1,v)$. It is the Cartesian product of a circle and a closed interval, and is homeomorphic to a closed annulus. A finite portion of the lateral surface of a circular cylinder realizes it in three-dimensional space. Its two boundary circles are $S^1\times \{-1\}$ and $S^1\times \{1\}$, and the matching seam directions make it an orientable surface. The Möbius band changes just one identification sign and loses both this two-component boundary and its orientability.

A probability measure-preserving transformation $T$ is weakly mixing when, for every pair of measurable sets $A,B$, the average absolute correlation discrepancy tends to zero:Equivalently, the correlations converge to their products in convergence in density of a sequence, its Cartesian product with itself is an ergodic transformation, or its product with every ergodic transformation is ergodic. The absolute values are essential: signed Cesaro convergence of a sequence of the correlations to their products characterizes ergodic transformations and does not imply weak mixing. For example, the transformation interchanging two equally weighted points has discrepancy $(-1{)}^n/4$ for a singleton $A=B$: its signed average tends to zero while its average absolute discrepancy stays $1/4$.