Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).

I need a reference to the following facts (which I believe are true at least in dimension $n=3$):

**Fact 1.** For every closed connected subset $A\subset X$ that can be embedded to $\mathbb R^{n-1}$ the complement $X\setminus A$ is connected.

**Fact 2.** For any closed subset $B\subset X$ whose connected components can be embedded to $\mathbb R$, the identity embedding $X\setminus B\to X$ induces an injective homomorphism $H_1(X\setminus B;G)\to H_1(X;G)$ in singular homologies with coefficients in some group $G$ (for example $\mathbb Z$ or $\mathbb Z/2\mathbb Z$).

**Remark.** The Alexander-Pontryagin Duality Theorem implies that Facts 1 and 2 are true if $X$ is the $n$-sphere. So, I need these facts for an arbitrary compact connnected $n$-manifold without boundary.

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