Differential Geometry – Can a Topological Manifold Be Non-Connected with Different Dimension Components

Question

These are two definitions in page 48 of the book an introduction to manifolds by Loring Tu.

Definition 5.1. A topological space $M$ is locally Euclidean of dimension $n$ if every point $p$ in $M$ has a neighborhood $U$ such that there is a homeomorphism $\phi$ from $U$ onto an open subset of $\mathbb R^n$.

Definition 5.2. A topological manifold is a Hausdorff, second countable, locally Euclidean space. It is said to be of dimension $n$ if it is locally Euclidean of dimension $n$.

In the last lines of page 48, we reed,

Of course, if a topological manifold has several connected components, it is possible for each component to have a different dimension.

But this is a bit strange for me. If a topological manifold has several connected components and each component has different dimension, then how this manifold can be locally Euclidean space, say for example of dimension $n$? That is, by the above definition of topological manifolad, can a non-connected toplogical space be a topological manifold?

Answer 1

As written, the term "locally Euclidean" is in fact not even defined at all (only "locally Euclidean of dimension $n$" is defined). What it appears the author really intended is the following pair of definitions:

Definition 5.1'. A topological space $M$ is locally Euclidean of dimension $n$ at a point $p\in M$ if $p$ has a neighborhood $U$ such that there is a homeomorphism $\phi$ from $U$ onto an open subset of $\mathbb R^n$. A topological space $M$ is locally Euclidean if for each $p\in M$, there exists $n$ such that $M$ is locally Euclidean of dimension $n$ at $p$.

Definition 5.2'. A topological manifold is a Hausdorff, second countable, locally Euclidean space. It is said to be of dimension $n$ if it is locally Euclidean of dimension $n$ at every point.

I would add, however, that this definition is not very standard. Most people define manifolds such that they must have the same dimension at every point, even if they are disconnected.