In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.

The definition of an atlas depends on the notion of a chart. A chart for a topological space M is a homeomorphism ${\displaystyle \varphi }$ from an open subset U of M to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair ${\displaystyle (U,\varphi )}$.

When a coordinate system is chosen in the Euclidean space, this defines coordinates on ${\displaystyle U}$: the coordinates of a point ${\displaystyle P}$ of ${\displaystyle U}$ are defined as the coordinates of ${\displaystyle \varphi (P).}$ The pair formed by a chart and such a coordinate system is called a local coordinate system, coordinate chart, coordinate patch, coordinate map, or local frame.

An atlas for a topological space ${\displaystyle M}$ is an indexed family ${\displaystyle \{(U_{\alpha },\varphi _{\alpha }):\alpha \in I\}}$ of charts on ${\displaystyle M}$ which covers ${\displaystyle M}$ (that is, ${\textstyle \bigcup _{\alpha \in I}U_{\alpha }=M}$). If for some fixed n, the image of each chart is an open subset of n-dimensional Euclidean space, then ${\displaystyle M}$ is said to be an n-dimensional manifold.

The plural of atlas is atlases, although some authors use atlantes.

An atlas ${\displaystyle \left(U_{i},\varphi _{i}\right)_{i\in I}}$ on an ${\displaystyle n}$-dimensional manifold ${\displaystyle M}$ is called an adequate atlas if the following conditions hold:^{[clarification needed]}

Every second-countable manifold admits an adequate atlas. Moreover, if ${\displaystyle {\mathcal {V}}=\left(V_{j}\right)_{j\in J}}$ is an open covering of the second-countable manifold ${\displaystyle M}$, then there is an adequate atlas ${\displaystyle \left(U_{i},\varphi _{i}\right)_{i\in I}}$ on ${\displaystyle M}$, such that ${\displaystyle \left(U_{i}\right)_{i\in I}}$ is a refinement of ${\displaystyle {\mathcal {V}}}$.

A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)