# Slice Chart Condition Proof - Topological Embedding

Below is some background information from Lee's Introduction To Smooth Manifolds about slice charts of embedded submanifolds. My question is at the bottom.

What I don't understand in the proof is the part highlighted in red. I don't understand how we can conclude that $$S$$ is a topological embedding from what we proved so far.

## 1 Answer

Up to the point you have highlighted, we have shown that for any point $$p$$ in $$S$$, we can define a chart $$(\psi,V)$$, where $$V$$ is open in $$S$$ and contains $$p$$, with $$\psi\colon V\to \widehat V\subset \Bbb R^k$$ a homeomorphism. The inclusion map $$\iota\colon S\hookrightarrow M$$ is a topological embedding because $$S$$ is given the subspace topology, hence $$\iota$$ is a homeomorphism onto its image.