The first surprise in manifold theory is that many spaces look Euclidean only when inspected through small windows. A circle is not an interval, a sphere is not a plane, and a torus cannot be flattened into the plane without cutting it. Yet near each point these spaces admit coordinates that make local questions resemble questions in $\mathbb R^n$. A topological manifold is the minimum topological framework in which this local Euclidean behaviour is made precise before any smooth, metric, or algebraic structure is added.

The definition is deliberately strict. Local Euclidean charts alone do not prevent global pathologies: a space can have local coordinates but fail to separate points well, or it can require too many coordinate patches to behave like an object of ordinary geometry. The Hausdorff and second countability hypotheses exclude these failures while preserving the familiar examples.

[example: A Circle Is Locally a Line] Let $S^1\subset\mathbb R^2$ be \begin{align*} S^1=\{(x,y)\in\mathbb R^2:x^2+y^2=1\}. \end{align*} Write $N=(0,1)$, and define stereographic projection from $N$ by sending a point $(x,y)\in S^1\setminus\{N\}$ to the intersection of the line through $N$ and $(x,y)$ with the horizontal axis. A point on that line has the form $(\lambda x,1+\lambda(y-1))$. Setting the second coordinate equal to $0$ gives $1+\lambda(y-1)=0$, so $\lambda=1/(1-y)$, and therefore \begin{align*} \sigma_N(x,y)=\frac{x}{1-y}. \end{align*} We compute its inverse. For $t\in\mathbb R$, the line through $N=(0,1)$ and $(t,0)$ is parametrized by $(\lambda t,1-\lambda)$. Its intersections with $S^1$ satisfy \begin{align*} (\lambda t)^2+(1-\lambda)^2=1. \end{align*} Expanding gives \begin{align*} \lambda^2t^2+1-2\lambda+\lambda^2=1. \end{align*} Subtracting $1$ and factoring gives \begin{align*} \lambda\bigl(\lambda(t^2+1)-2\bigr)=0. \end{align*} The solution $\lambda=0$ gives the north pole, so the other intersection is obtained from $\lambda=2/(t^2+1)$. Thus \begin{align*} \sigma_N^{-1}(t)=\left(\frac{2t}{t^2+1},\frac{t^2-1}{t^2+1}\right). \end{align*} This point lies on $S^1$ because \begin{align*} \left(\frac{2t}{t^2+1}\right)^2+\left(\frac{t^2-1}{t^2+1}\right)^2=\frac{4t^2+t^4-2t^2+1}{(t^2+1)^2}=\frac{t^4+2t^2+1}{(t^2+1)^2}=1. \end{align*} Also, \begin{align*} 1-\frac{t^2-1}{t^2+1}=\frac{2}{t^2+1}, \end{align*} so applying $\sigma_N$ to this point gives \begin{align*} \frac{2t/(t^2+1)}{2/(t^2+1)}=t. \end{align*} The formulas for $\sigma_N$ and $\sigma_N^{-1}$ are continuous, so $\sigma_N:S^1\setminus\{N\}\to\mathbb R$ is a homeomorphism. Rotating the circle moves any chosen point to $N$, so every point of $S^1$ has a neighbourhood with one-dimensional coordinates, even though the whole circle is compact and no nonempty open subset of $\mathbb R$ is compact. [/example]

This example shows the guiding tension. Local coordinates make the space look linear, while global topology records how those coordinate patches are glued together. The subject begins by separating those two roles.

## Definition

### The Manifold Axioms

The main definition has to say exactly which topological spaces are eligible for coordinate-based geometry. Local Euclidean behaviour supplies the small windows, but the Hausdorff and countability assumptions keep those windows from being glued into doubled points or unmanageably large topologies. The definition below records both the local Euclidean requirement and the global topological discipline needed for geometry.

[definition: Topological Manifold] Let $n$ be a fixed nonnegative integer. A topological manifold of dimension $n$ is a Hausdorff, second countable [topological space](/page/Topological%20Space) $M$ such that for every point $p\in M$ there exist an [open set](/page/Open%20Set) $U\subset M$ with $p\in U$, an open set $W\subset\mathbb R^n$, and a homeomorphism \begin{align*} \varphi:U&\to W. \end{align*} [/definition]

The first condition says that distinct points can be separated by disjoint open neighbourhoods; the second says that the topology has a countable basis; and the third says that $M$ is locally homeomorphic to $\mathbb R^n$. The integer $n$ is the dimension of the manifold, but a single chart is rarely enough to describe all of $M$.

### Charts and Atlases

To make the phrase "locally Euclidean" usable in computations, the coordinate neighbourhood must be open in the space and must map onto an open subset of Euclidean space. The openness condition lets local topological statements transfer across the coordinate map without changing their nature.

[definition: Coordinate Chart] Let $M$ be a topological manifold of dimension $n$. A coordinate chart on $M$ is a pair $(U,\varphi)$ such that $U\subset M$ is open and there is an open set $W\subset\mathbb R^n$ for which \begin{align*} \varphi:U&\to W \end{align*} is a homeomorphism. [/definition]

The coordinates of a point $p\in U$ are the [real numbers](/page/Real%20Numbers) $(x_1,\ldots,x_n)=\varphi(p)$, but at this stage the coordinates only preserve topology. They do not yet allow derivatives, lengths, angles, or integration.

To use coordinates in practice, the coordinate neighbourhoods must be organized as a reusable cover of the whole space. This need leads to the atlas, the object that isolates the coordinate system as a collection rather than a single chart. Most manifolds cannot be covered by one chart, and the data of many overlapping charts becomes the language in which global questions are translated into Euclidean ones.

[definition: Atlas] Let $M$ be a topological manifold of dimension $n$. An atlas on $M$ is a collection of coordinate charts $\{(U_i,\varphi_i)\}_{i\in I}$ such that \begin{align*} M=\bigcup_{i\in I}U_i. \end{align*} [/definition]

An atlas is a topological object here. Later, when smooth manifolds are introduced, the transition maps between overlapping charts must satisfy differentiability conditions. For a topological manifold, only continuity and continuous invertibility are required.

## Local Models and Dimension

### Dimension Invariance

The definition requires a fixed integer $n$. This is not a cosmetic choice. If different points could have different Euclidean dimensions, the space would not have a stable local type, and the usual geometric constructions would not have a single rank.

[quotetheorem:9910]