I am reading about topological manifolds from An Introduction to Manifolds by Loring Tu (Second Edition, page no. 48). He defines topological manifolds as follows.
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 next paragraph, he says,
Of course, if a topological manifold has several connected components, it is possible for each
component to have a different dimension.
I understand what a component of a topological space is and also know that a component is connected as well as closed in it. But I don't understand what the dimension of a component is.
My Question: What is the definition of dimension of a component of a topological space? How can we determine the dimension? Can you provide a simple example to illustrate it?
Best Answer
In 5.1 the $n$ is part of the definition, so all points have neighbourhoods like open sets in $\Bbb R^n$.
In 5.2. the definition is more general: every point has its own local dimension $n(x)$, as it were, and then it's a simple theorem that in each connected component of $X$, the same $n$ is used for all its points. So in that definition a disjoint union of a circle and a sphere would be valid, and in 5.1 it wouldn't be.