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I thought of this weird and really simple folklore in elementary topology based on the equivalence of the definitions of regular submanifold and embedded submanifold, from elementary differential geometry, if it's correct.

Is this correct? If so, then know any books/references that say this explicitly?

Let $B$ be a topological space and $A$ a subset of $B$ (with subspace topology). The inclusion map $\iota: A \to B$ is an open map if and only if $A$ is open in $B$.

Note: I assume the above is correct in attempting to answer this question: Are these equivalent for regular submanifold? Open, image of local diffeomorphism, image of injective local diffeomorphism

BCLC
  • 13,459

1 Answers1

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  1. Yes. Here's the proof:

Only if direction: $\iota(V)=V$ is open in $B$ for all $V$ open in $A$. Choose $V=A$ itself.

If direction: Let $V$ open in $A$. We must show that $\iota(V)$ is open in $B$. Because $A$ is open in $B$, we have $V$ open in $B$. Finally, $\iota(V)=V$.

  1. Closest I could find is:

Properties of inclusion map between topological spaces.

BCLC
  • 13,459