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  1. In some textbooks, not all manifolds have (uniform) dimension, so $I = (0,1) \cup \{2\}$ might be considered a dimensionless manifold in the sense that $I$ is not an $n$-manifold for any integer $n$ but is a manifold with pointwise dimensions $n_p=1$ for each $p \in C_1 := (0,1)$ and $n_q=0$ for each $q \in C_2 := \{2\}$, where we observe that $C_1$ and $C_2$ are the connected components of $I$.

    • Therefore, manifolds (without boundary) with dimensions need not unite to a manifold (without boundary) with dimensions but can still unite to a manifold (without boundary) (namely without uniform dimension).

  2. We can also have manifolds (without boundary) with dimensions unite to manifolds with boundary with or without dimensions such as $N=\{0,2\} \cup (0,1) = [0,1) \cup \{2\}$ and $D^2 = B^2 \cup S^1$ where $D^2$ and $B^2$ are the, respectively, closed and open unit disks with unit circle $S^1$ as their topological boundary in $\mathbb R^2$.

    • Here, $D^2$ has dimension 2, while $N$ is (uniform) dimensionless.

  3. Of course we can have $D = \{0,2\} = \{0\} \cup \{2\}$, a manifold (without boundary) with dimension, namely the same dimension as each of its connected components.

What are some examples where 2 manifolds without boundaries but with dimensions unite to something other than in the 3 situations above?

BCLC
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    The two manifolds may intersect non-trivially. Consider a sphere and a plane (like the plane of the equator) in 3D. With more complicated shapes the possibilities are very diverse. For example, there is no bound to the number of components of the intersection. – Jyrki Lahtonen Sep 11 '19 at 04:39
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    Or, if you prefer 1-dimensional manifolds, consider the union of $y=\sin x$ and $y=\cos x$, Or the union of the $x$-axes and the graph of a polynomial with $n$ zeros. – Jyrki Lahtonen Sep 11 '19 at 05:25
  • @JyrkiLahtonen Thanks. Could those unions be immersed submanifolds with or without boundary of the same manifolds, which those manifolds may be considered submanifolds of? For example, $\sin x$ and $\cos x$ are submanifolds of $\mathbb R^2$, so for the union of $\sin x$ and $\cos x$, if I ask if this union is an immersed submanifold with or without boundary, then what I mean is to ask if the union would be an immersed submanifold with or without boundary of $\mathbb R^2$. –  Sep 15 '19 at 12:29

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