If $\varphi: A\to X$ is injective and continuous such that $\tilde \varphi: A\to \varphi(A)$ is a homeomorphism, is $\varphi(A)$ open in $X$?

Question

Let $(A,T)$ and $(X,T')$ two topological spaces. Suppose that $$\varphi: A\to X$$ is continuous and injective and that $$\tilde \varphi: A\to \varphi(A)$$

is a homeomorphism.

1) Is $\varphi(A)$ open in $X$ ?

2) If no, if $A\subset X$ does $\varphi(A)$ open in $X$ ?

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I know that for $X=\mathbb R^n$ and $A\subset \mathbb R^n$ it's true and it's call "Brouwer invariance domain". Does it still hold in unspecific topological spaces ? If no, do you have a counter example ?

Answer 1

No, take $X=\mathbb{R}$ and $A$ the point $\{0\}$. Consider the canonical embedding.

Answer 2

No in both cases, for any subspace $A \subseteq X$ , the canonical embedding $i: A \to X$ defined by $i(x) = x$ is continuous, injective and $i:A \to i[A]$ is by definition (of the subspace topology) a homeomorphism. But not all subspaces are open...