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The intrinsic topology of an immersed submanifold need not be the subspace topology
Statement
False claim: an immersed submanifold always carries the subspace topology of its image in the ambient manifold.
Facts & Assumptions
Given: The same componentwise inclusion from the countable family of concentric circles used above.
An immersed submanifold is a manifold with an injective immersion into the ambient manifold; its intrinsic topology is not defined to be the subspace topology (Immersed submanifolds).
Embedded submanifolds, by contrast, do use the subspace topology (Embedded submanifolds and slice charts).
The disjoint-union source is a smooth manifold and each circle is smooth (Countable disjoint unions of fixed-dimensional smooth manifolds are smooth manifolds, A regular level set is an embedded submanifold, For a natural the function is differentiable everywhere with derivative ; for it is the constant , with derivative ; for a natural the function is differentiable at every with derivative ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Sums, scalar multiples, products and quotients: , , , and when ).
Refutation
By [L1], the source is a smooth one-manifold, and the map is an injective immersion componentwise. Hence is an immersed submanifold in the sense of [F1].
In the intrinsic topology of , each circle component is open because is a disjoint union. In the subspace topology on , the unit circle component is not open because every neighbourhood of one of its points meets infinitely many outer circles. Thus the two topologies differ.
Therefore an immersed submanifold need not carry the subspace topology of its image.
Depends on
- Embedded submanifolds and slice charts
- Immersed submanifolds
- Countable disjoint unions of fixed-dimensional smooth manifolds are smooth manifolds
- A regular level set is an embedded submanifold
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
Used by
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Sources
- John M. Lee, Introduction to Smooth Manifolds, Immersed Submanifolds (standard reference, not scraped)