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Every set lying between a connected set and its closure is connected.

An arbitrary product topology of connected spaces is connected. For two nonempty factors, fix $y_0\in Y$ and use the connected subsetsThey cover $X\times Y$ and share the connected horizontal slice. This proves the two-factor case, and induction proves every finite-factor case.

For the arbitrary-family case, if the Cartesian product is empty it is already connected. Otherwise fix one tuple $a$. For every finite index set $F$, let $X_F$ consist of tuples that equal $a$ outside $F$. Each $X_F$ is a finite product of connected spaces, and all contain $a$, so their union $D$ is connected. Every nonempty basic open set restricts only finitely many coordinates, and therefore meets $D$. Thus $D$ is dense, and the closure of a connected set proves that the whole Cartesian product is connected.

If connected subsets cover a space and every two of them intersect, then their union is connected. Equivalently, any continuous integer-valued function is constant on each member, and the intersections force all those constants to agree.

A topological space is connected exactly when every continuous map from it to the discrete space $\mathbb{Z}$ is constant. A disconnection produces a nonconstant indicator map, while the continuous image of a connected space in $\mathbb{Z}$ must be a singleton.

If $f:X\to Y$ is a continuous map and $X$ is a connected space, then its image $f(X)$, with the subspace topology from $Y$, is a connected subset of $Y$. The whole codomain $Y$ need not be connected; the assertion concerns the values actually attained. No injectivity is required.^[ref]

For example, the continuous function $f(x)=x^2$ maps the connected closed interval $[-2,2]$ onto the connected closed interval $[0,4]$, even though opposite inputs have the same output. The same continuous function maps the disconnected union $[-2,-1]\cup [1,2]$ onto $[1,4]$. Thus a connected image does not imply a connected source: a continuous map can merge different connected components.

For a continuous function $r:[0,1]\to \mathbb{R}$ representing a continuous reading, $r(0)=0$ and $r(1)=1$ force some input with $r(t)=1/2$. Its image is a connected subset of the real line, hence a real interval containing both endpoint readings. For the synthetic model $r(t)=t^2$, the missing-value test is explicit: $r(1/\sqrt{2})=1/2$. A step reading equal to $0$ for $t<1/2$ and $1$ for $t\ge 1/2$ skips $1/2$, but fails continuity at $t=1/2$.

Proof outline: a separation of a topological space of $f(X)$ into nonempty open subsets pulls back under $f$ to nonempty open subsets of $X$. Their intersection is empty and their union is $X$, contradicting connectedness. This also shows why a continuous map sends each connected component of its source into a single connected component of its target.

A connected component of a topological space $X$ is a nonempty connected subset that is maximal under subset inclusion. If $C$ is a connected component and $C\subseteq D\subseteq X$ with $D$ a connected subset, then $D=C$. Maximality means that no larger connected subset contains $C$; it does not mean largest cardinality among all connected subsets.

For example, the connected components of $[-2,2]\setminus \{0\}$ in the usual subspace topology of the real line are $[-2,0)$ and $(0,2]$. Each is a real interval, and no connected subset of the real line can meet both without containing the missing point $0$. For the finite set $\{-2,0,2\}$ in the same subspace topology, the three connected components are its singleton sets.

Every point $x\in X$ lies in exactly one connected component, constructed by taking the unionAll members contain $x$, so their union is connected by connected union with a connected hub. The union includes every connected subset through $x$, so it is maximal. Equivalently, defineThis is an equivalence relation: singleton sets give reflexivity, symmetry is immediate, and two connected subsets sharing the middle point have a connected union, giving transitivity. The equivalence classes are precisely the connected components. They form a partition of a set: they cover $X$ and any two are either equal or disjoint.^[ref]

Every nonempty connected subset lies within one connected component. Moreover, every connected component is closed in $X$: closure of a connected set makes its closure connected, and maximality forces that closure to equal the original connected component. A connected component need not be open. For example, in $S=\{0\}\cup \{1/n:n=1,2,\dots \}$ with its usual subspace topology from the real line, all connected components are singleton sets, but $\{0\}$ is not open: every neighbourhood of $0$ in $S$ contains some $1/n$.^[ref]

The map $z\mapsto \cosh (\pi z)$ is a conformal bijection fromonto the upper half-plane. Its three boundary intervals map respectively to $(1,\infty )$, $(-1,1)$ and $(-\infty ,-1)$.

If a holomorphic function $f$ satisfies $f^{\prime }(z_0)\ne 0$, then it has a holomorphic local inverse near $z_0$ and is conformal there. Its possible failures of local conformality are therefore its critical points.

Two plane domains are conformally equivalent when there is a bijective holomorphic map between them whose inverse is holomorphic.

In the simply typed lambda calculus, types are generated from base types by the arrow constructor $A\to B$. A context $\Gamma$ assigns types to variables. Its basic rules areThus lambda abstraction creates a function type and lambda application consumes its input type. For example, $\lambda x.x$ has type $A\to A$, and $\lambda f.\lambda x.fx$ has type $(A\to B)\to A\to B$.

The untyped self-application $xx$ would require one finite type to satisfy $A=A\to B$, which is impossible in this type grammar. This excludes the familiar looping term $(\lambda x.xx)(\lambda x.xx)$; more generally the Strong normalization theorem for simply typed lambda calculus says that every reduction sequence from a well-typed term terminates. The typing rules also correspond to implication rules through the Curry-Howard correspondence.

Disjoint sets $A,B\subseteq \mathbb{N}$ are recursively inseparable when no recursive set $C$ satisfies $A\subseteq C$ and $B\cap C=\varnothing$.

The arithmetical hierarchy measures how many alternating blocks of unbounded quantifiers over natural numbers are needed to define a set of natural numbers. In its lightface version, $A\in {\Sigma }_n^0$ when membership has the formwith a computable predicate $R$ and $n$ alternating blocks; a block can code a finite tuple. The class ${\Pi }_n^0$ starts with a universal block, and ${\Delta }_n^0={\Sigma }_n^0\cap {\Pi }_n^0$. At level zero the matrix is computable; these definitions allow natural-number parameters, rather than arbitrary sets as oracle parameters.

The halting set is ${\Sigma }_1^0$: membership says that some finite computation halts. Its complement is ${\Pi }_1^0$. The index set of total computable functions is ${\Pi }_2^0$, since totality says $\forall x\exists s$ that the computation on input $x$ halts within $s$ steps. Alternation therefore records a concrete difference between witnessing one successful finite search and certifying that every input eventually succeeds.

Dovetailing interleaves finite stages of multiple computations so that every computation that halts is eventually observed.

For a standard effective numbering $e\mapsto {\phi }_e$ of partial computable functions, there is a total computable function $p(e,k)$, injective in the pair $(e,k)$, such that ${\phi }_{p(e,k)}={\phi }_e$. Thus infinitely many different program indices can describe exactly the same computation. The uniformity matters: the extra index can be chosen algorithmically, rather than merely known to exist.

For a concrete Turing machine coding, insert an unreachable block carrying the original description and the integer $k$, with a recognizable delimiter. Execution never enters that block, so the partial computable function remains unchanged; the recognizable description makes the padded indices distinct. The general version for acceptable numberings transfers this construction through a computable isomorphism of numberings. Padding distinguishes properties of program indices from properties of the partial computable functions they compute.

An index set is a set of program codes whose membership depends only on the partial function or enumerable language computed, rather than the code's syntax.

A many-one reduction $A{\le }_{m}B$ is a total computable $f$ satisfying $x\in A$ exactly when $f(x)\in B$.

A set is computably enumerable when a program halts precisely on its members. Equivalently, it is the domain of a partial computable function or some algorithm lists its elements.

A list $f(n,k)$ of all functions in a class closed under diagonal substitution and successor misses $d(k)=f(k,k)+1$.

Primitive recursive functions are generated by zero, successor, and projections under composition and primitive recursion.