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Let $S^{-1}:\Sigma \to \Lambda$ be inverse of itinerary function. I showed that $S$ is continuous and bijective. How to show that $S^{-1}$ is continuous?

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    See for information, it is kinda long definition https://docs.google.com/viewer?a=v&q=cache:epRwB-oNlkEJ:ocw.mit.edu/courses/mathematics/18-091-mathematical-exposition-spring-2005/lecture-notes/lecture06.pdf+itinerary+function+is+homeomorphism+proof&hl=ru&gl=ru&pid=bl&srcid=ADGEESimn5hLGdsKAiFTYMQwBx8a9kQi_CTUqQdRIHSRuEYxhu-Y7AR4GEiQGcnQuVngvPVkZdPgJOxY6cSBKUfrS0ZMlAgm-SwKNiHEyqVkH9upb6eOCdRqg_R45ifCue-pHvkQ1rae&sig=AHIEtbRK28HC-poG80TpFIOHEzhxCBJDMg2012-11-07

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Showing that $S^{-1}$ is continuous is the same as showing that $S$ is a closed map. For this, we can use the oft-cited result from general topology:

Let $f\colon X\to Y$ be a continuous bijection from a compact space $X$ to a Hausdorff space $Y$. Then $f$ is closed, and hence a homeomorphism.

The proof of this is simple. If $E\subseteq X$ is closed, then $E$ is compact since $X$ is compact. Since $f$ is continuous, it follows that $f(E)$ is compact. Since compact subsets of Hausdorff spaces are closed, it follows that $f(E)$ is closed.