Countable base Banach manifold and smooth map

Let $k\in \mathbb{N}\cup \{\mathrm{\infty }\}$ and let $E$ be a real Banach space (Banach space) with its norm topology (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

• A chart on a topological space $M$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) is a pair $(\phi ,U)$ in which $U\subseteq M$ is open and $\phi :U\to \phi [U]\subseteq E$ is a homeomorphism onto an open subset of $E$ (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological); $U$ is the domain of the chart and $\phi$ its coordinate map.
• Two charts $(\phi ,U)$ and $(\psi ,V)$ of $M$ with $U\cap V\mathrm{\ne }\mathrm{\varnothing }$ are $C^k$ compatible when the transition map $\psi \circ {\phi }^{-1}:\phi [U\cap V]\to \psi [U\cap V]$ and its inverse are of class $C^k$ (C k map between Banach spaces) as maps between open subsets of $E$. Two charts with disjoint domains are declared compatible.
• An atlas of class $C^k$ on $M$ is a family of pairwise $C^k$ compatible charts whose domains cover $M$, and the space $M$ together with such an atlas is a $C^k$ Banach manifold modelled on $E$ — a Banach manifold for short — when additionally $M$ is Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) and second countable (Second countability: an at most countable basis for the topology).

Thus every point of $M$ lies in the domain of a chart: $M$ is locally homeomorphic to open subsets of the Banach space $E$, and the change of coordinates between any two charts in its specified atlas is a $C^k$ map. Henceforth a chart of the structured manifold $M$ means a member of that specified atlas; it does not mean an arbitrary local homeomorphism on the underlying topological space. The pair (Hausdorff, second countable) is part of the definition and is never dropped below.

Let $M$ be a $C^k$ Banach manifold modelled on $E$ and $N$ a $C^k$ Banach manifold modelled on $F$, and let $f:M\to N$ be a map.

• $f$ is of class $C^k$ when for every chart $(\phi ,U)$ in the specified atlas of $M$ and every chart $(\psi ,V)$ in the specified atlas of $N$ the coordinate representative $\psi \circ f\circ {\phi }^{-1}:\phi [U\cap f^{-1}[V]]\to F$ has open domain $\phi [U\cap f^{-1}[V]]\subseteq E$ and is of class $C^k$ on that domain. Openness is part of the requirement, not an assumption about an arbitrary set map $f$.
• $f$ is smooth when it is of class $C^{\mathrm{\infty }}$.
• A bijection $f:M\to N$ is a $C^k$ diffeomorphism when both $f$ and $f^{-1}$ are of class $C^k$; likewise for $C^{\mathrm{\infty }}$.

• Chart independence of the class of a map is a chain-rule statement. If $(\phi ,U)$, $({\phi }^{\mathrm{\prime }},U^{\mathrm{\prime }})$ are charts of $M$ and $(\psi ,V)$, $({\psi }^{\mathrm{\prime }},V^{\mathrm{\prime }})$ are charts of $N$ from their specified atlases, then on the open set where both sides are defined, ${\psi }^{\mathrm{\prime }}\circ f\circ {\phi }^{\mathrm{\prime }-1}$ equals $({\psi }^{\mathrm{\prime }}\circ {\psi }^{-1})\circ (\psi \circ f\circ {\phi }^{-1})\circ (\phi \circ {\phi }^{\mathrm{\prime }-1})$, a composite of $C^k$ transition maps and the representative $\psi \circ f\circ {\phi }^{-1}$. For $k=1$ the class is therefore independent of the charts by the chain rule (Chain sum product and composition rules for Banach derivatives) — the composition of $C^1$ maps is $C^1$ because the chain rule expresses the derivative as a product of continuous operator-valued maps (C k map between Banach spaces) — and consequently $C^1$-ness may be checked at each point with one pair of charts around it. Nothing below uses this independence for $k\ge 2$, and the definition itself quantifies over all pairs from the specified atlases, so no higher-order chain rule is presupposed.

• The model space is fixed. Charts take values in one Banach space $E$, which may be infinite dimensional; a manifold with a finite-dimensional model space is the familiar finite-dimensional case. Two manifolds modelled on different Banach spaces are compared by maps whose coordinate representatives map open subsets of one model space into the other.

• Open sets are the basic examples, when the model space is second countable. If $E$ is second countable, then an open subset $W$ of $E$ is a $C^{\mathrm{\infty }}$ Banach manifold modelled on $E$ with the single chart $({\mathrm{i}\mathrm{d}}_W,W)$: a basis of $E$ restricts to a basis of the subspace $W$, and $W$ is Hausdorff because $E$ is. The hypothesis cannot be dropped: for $E={\mathrm{\ell }}^{\mathrm{\infty }}$ and $W=E$ the identity chart covers $E$, but $E$ is not second countable — the uncountably many $0$-$1$ sequences are pairwise at distance $1$. Their radius-$1\mathrm{/}3$ balls are pairwise disjoint. A countable basis would assign to each such sequence the least indexed basis member containing it and contained in its ball, giving an injection of the uncountable set of $0$-$1$ sequences into $\mathbb{N}$, a contradiction. A map between open subsets of a second countable $E$ is of class $C^k$ as a map of manifolds exactly when it is of class $C^k$ in the sense of C k map between Banach spaces. All the local theorems of this page are statements about such open sets, transported to manifolds exactly through charts.

• The Hausdorff and countability hypotheses are part of the definition. They are the standard hypotheses of the global theory: the countable base is what the later Sard–Smale and transversality development consumes, and the Hausdorff condition is what makes the local pieces of a manifold fit together as a space of points rather than a set with overlapping coordinate patches. No theorem on this page asserts anything for a non-Hausdorff or non-second-countable "manifold", and the four Euclidean-space local theorems above are unaffected by either hypothesis because they do not mention manifolds at all.