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For a continuous map $f:M\to N$ of nonzero mapping degree between closed connected oriented manifolds of the same dimension, $f^{\ast }:H^q(N;\mathbb{Q})\to H^q(M;\mathbb{Q})$ is injective in every degree. Pair a nonzero class $a$ with a complementary class $b$ using Poincare duality. Naturality gives $⟨f^{\ast }a⌣f^{\ast }b,[M]⟩=(\mathrm{deg}f)⟨a⌣b,[N]⟩\ne 0$, so $f^{\ast }a\ne 0$. Over coefficients of positive characteristic the argument requires that the degree remain nonzero in that field.

Let $X$ be a CW complex of dimension at most $n$, with $n\ge 1$. Every class $a\in H^n(X;\mathbb{Z})$ is $f^{\ast }u$ for a continuous map $f:X\to S^n$ and an orientation generator $u\in H^n(S^n;\mathbb{Z})$.

Choose a cellular cocycle representing $a$ and collapse the $(n-1)$-skeleton to a point. The quotient is a wedge sum of one $n$-dimensional sphere for every $n$-cell. map each sphere to $S^n$ with mapping degree equal to the chosen cellular coefficient. Composing with the quotient gives the required continuous map. There are no higher-dimensional cells whose attachments could obstruct it; the weak topology of a CW complex also permits an infinite wedge.

The restriction $n\ge 1$ is essential. On a three-point discrete space, an integral degree-zero cohomology class can take the three values $0,1,2$. pullback along a map to $S^0$ can take at most the two values of the chosen class on $S^0$, so it cannot realize that class.

The closed unit ball $B$ in the infinite-dimensional real Hilbert space ${\ell }^2$ admits a retraction onto its unit sphere. Write $S$ for the forward unilateral shift operator and set $F(x)=\sqrt{1-\Vert x{\Vert }^2}{e}_0+Sx$. Then $\Vert F(x)\Vert =1$ and $F(x)\ne x$: equality would force all coordinates of $x$ to coincide, hence vanish, contradicting $F(0)=e_0$.

For $d=x-F(x)$, setThe nonvanishing denominator makes $r$ a continuous map. Expanding its squared norm gives $\Vert r(x)\Vert =1$. On $\Vert x\Vert =1$, the fraction is one, so $r(x)=x$. Thus the finite-dimensional no-retraction theorem does not extend to this closed unit ball.

A right-topological semigroup is a semigroup with a topology such that, for each fixed $a$, the right translation $R_a:x\mapsto xa$ is a continuous map. The varying input is on the left. Requiring $x\mapsto ax$ instead gives a left-topological semigroup; requiring multiplication jointly continuous gives a topological semigroup.

For a one-sided example, take all functions $\mathbb{R}\to \mathbb{R}$ with function composition and the product topology, so convergence is pointwise convergence. For fixed $g$, the map $f\mapsto f\circ g$ is continuous because its value at $x$ is the evaluation of $f$ at the fixed point $g(x)$.

Left composition can fail to be continuous. Let $g(x)=1$ for $x>0$ and zero otherwise, and let $f_n$ be constant $1/n$. Then $f_n$ converges pointwise to the zero function, but $g\circ f_n$ is constantly one while $g\circ 0$ is constantly zero. Thus right-translation continuity alone does not imply continuity of the other translations.

For based continuous maps $\alpha :S^p\to G$ and $\beta :S^q\to G$ into a topological group, take the pointwise commutatorIt is exactly the identity on $S^p\vee S^q\subseteq S^p\times S^q$, so it descends to the smash product $S^p\wedge S^q\cong S^{p+q}$. Its homotopy class is the Samelson product $⟨\alpha ,\beta ⟩\in {\pi }_{p+q}(G)$. This measures failure of the two maps to commute; it vanishes for an abelian group.^[ref]

For an ordinary loop space, reversal is an inverse up to homotopy, and a retraced loop need not be the constant loop. Literal descent therefore requires a strict topological group model or the corresponding coherent contractions, rather than treating loop reversal as a strict inverse. Equivalently, suspension-loop adjunction defines the operation through the sign-adjusted Whitehead product in the original space.

In positive degrees these operations give the graded antisymmetry and Jacobi identity of the loop-space homotopy Lie algebra. Under the Hurewicz homomorphism, the product becomes the graded commutator in the Pontryagin ring.

Samuel Eilenberg was a mathematician whose work connected algebraic topology, homological algebra and category theory. With Saunders Mac Lane, he introduced the language of categories, functors and natural transformations; with Norman Steenrod, he gave an axiomatic account of homology and cohomology.^[ref] This language records more than the invariants attached to spaces: a continuous map also induces a map between the corresponding algebraic objects. Tracking those induced maps makes it possible to compare constructions and explain why many apparently different versions of a theory express the same underlying structure.

The continuous map projecting a real number onto the closed interval $[a,b]$, with $a\le b$. It equals $a$ below the interval, equals the input inside it, and equals $b$ above it. It is nondecreasing and has Lipschitz constant one.

Let $K$ be a nonempty compact convex set in a Banach space and $T:K\to K$ a continuous map. For every $\epsilon >0$, there is a finite-dimensional linear subspace $L_{\epsilon }$ and a continuous map $T_{\epsilon }:K\to K\cap L_{\epsilon }$ that approximates $T$ uniformly to within $\epsilon$ in norm.

Choose $y_1,\dots ,y_N\in T(K)$ whose open radius-$\epsilon$ balls cover the compact image $T(K)$. Set $w_i(x)=\max \{0,\epsilon -\Vert Tx-y_i\Vert \}$ andThe denominator is positive. This convex combination belongs to $K\cap \mathrm{span}\{y_1,\dots ,y_N\}$ and depends continuously on $x$. Every contributing $y_i$ has distance less than $\epsilon$ from $Tx$, so the triangle inequality gives $\Vert T_{\epsilon }(x)-Tx\Vert <\epsilon$.

Every continuous map from a nonempty compact convex set $K$ in a Banach space to itself has a fixed point. A useful equivalent form takes a nonempty closed bounded convex set $C$ and a continuous map $T:C\to C$ whose image is relatively compact. To obtain this version from the compact-domain version, restrict $T$ to $\overline{\mathrm{conv}}T(C)\subset C$ and use closed convex hull of a compact set.

For a proof on $K$, the Schauder approximation lemma gives continuous maps $T_{\epsilon }:K\to K\cap L_{\epsilon }$ with finite-dimensional $L_{\epsilon }$ and $\Vert T-T_{\epsilon }{\Vert }_{\infty }\le \epsilon$. The Brouwer fixed-point theorem on the nonempty compact convex set $K\cap L_{\epsilon }$ gives $x_{\epsilon }=T_{\epsilon }{x}_{\epsilon }$, so $\Vert T{x}_{\epsilon }-x_{\epsilon }\Vert \le \epsilon$. A convergent subsequence as $\epsilon \downarrow 0$ and the continuity of $T$ give an exact fixed point.

A singular $n$-simplex in a space $X$ is a continuous map $\sigma :{\Delta }^n\to X$ from the standard $n$-simplex. Its alternating restrictions to the faces form its boundary.

A continuous map from a smooth manifold to ${\mathbb{R}}^d$ can be approximated by a smooth map with any prescribed positive continuous pointwise error bound. In coordinate charts, smoothing by local convolution and then combining with a smooth partition of unity gives the approximation. On a compact subset of an open chart domain, a smooth cutoff allows the resulting straight-line homotopy to be supported in that domain. This permits the Sard theorem to be used after approximation; the original continuous map can be space-filling and need not omit a point.

Every integer $m$ is the mapping degree of a continuous map $S^n\to S^n$ in every dimension $n\ge 1$. Start with the circle map $z\mapsto z^m$, whose argument winds $m$ times, and use degree under suspension repeatedly. Negative $m$ reverses the winding, and $m=0$ is constant before suspending. The degrees of maps of the zero-sphere are a genuine exceptional case.

For a Tychonoff space $X$, its Stone-Čech compactification $\beta X$ has the following universal property: every continuous map $f:X\to K$ into a compact Hausdorff space extends to a continuous map $\overline{f}:\beta X\to K$. The original space is embedded as a dense subset of $\beta X$. The hypothesis on $X$ matters because a general space need not embed in any compact Hausdorff space.

The extension is unique: two continuous maps into a Hausdorff space that agree on a dense subset agree everywhere. If two Hausdorff compactifications have this extension property, applying it to the inclusion of $X$ into each one gives comparison maps in both directions. Their composites fix the dense subset $X$, so uniqueness makes both composites identity maps. The comparison maps are therefore inverse homeomorphisms fixing $X$. Thus the universal property characterizes the Stone-Čech compactification up to equivalence, rather than depending on a particular construction.

A standard form of the Sullivan conjecture, proved by Haynes Miller, states that for a finite group $G$ and a connected finite-dimensional CW complex $X$, the pointed mapping space from the classifying space of a discrete group $BG$ to $X$ is a weakly contractible space:Thus every basepoint-preserving continuous map $BG\to X$ is null-homotopic, and higher homotopy families of such maps are trivial as well. Miller's Theorem A allows the more general assumption that every finitely generated subgroup of $G$ is finite.^[ref]

The finite-dimensional target hypothesis is substantial. For $G=\mathbb{Z}/2\mathbb{Z}$, maps from $BG$ to a finite-dimensional sphere satisfy the theorem, while the identity map of $BG$ is not null-homotopic: it induces the identity on its nontrivial fundamental group. That target $BG$ is infinite-dimensional. The theorem concerns these unstable mapping spaces; it cannot simply be transferred to every stable mapping construction.

If a continuous map $g:{\mathbb{R}}^n\to {\mathbb{R}}^n$ has $\Vert g(x)-x\Vert \le K$, then it is onto. For a target $y$, choose $R\ge K+\Vert y\Vert$, $R>0$, and map the unit ball to itself by $f(x)=x-(g(Rx)-y)/R$. A fixed point gives $g(Rx)=y$.

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.

For a continuous map $f:M\to N$ between connected closed oriented manifolds of the same positive dimension $n$, the topological degree is the integer determined byThe chosen fundamental classes fix the signs. This homology definition makes sense without differentiability and is invariant under homotopy. For spheres, it is the integer by which the induced map multiplies their top homology group.^[ref]

If $f$ is smooth and $y$ is a regular value, the finite preimage giveswhere the sign compares the chosen orientations in source and target. Thus opposite local signs can cancel: topological degree is a signed count, rather than the number of preimages.

For the circle map $z\mapsto z^k$, with $|z|=1$ and $k$ an integer, the topological degree is $k$. An orientation-reversing reflection has topological degree $-1$, while a constant map has topological degree zero. A nonzero topological degree forces surjectivity; a homotopy between continuous maps preserves their topological degree.

Topological dynamics studies continuous maps and their orbits on topological spaces. For a compact metric space $X$ and a continuous map $T:X\to X$, the pair $(X,T)$ is a compact Hausdorff topological dynamical system.

A continuous map injective map $f:X\to Y$ that is a homeomorphism onto its image with the subspace topology. A continuous map injective map from a compact space into a Hausdorff space is a topological embedding.

Topological entropy measures the exponential growth of distinguishable orbit histories. For a continuous map $F$ on a compact metric space, compare two initial states by the largest distance between their first $n$ iterates:Let $s_n(ϵ)$ be the largest number of initial states whose histories are pairwise more than $ϵ$ apart in this metric. ThenThis is a count at fixed observational resolution before the resolution is refined.^[ref]

The identity map has zero topological entropy: observing longer does not distinguish additional histories at fixed resolution. In the binary symbolic dynamics of a full two-symbol shift, there are $2^n$ length-$n$ words, giving entropy $\log 2$. Thus positive topological entropy expresses persistent exponential orbit complexity, rather than merely a large instantaneous displacement.