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Let \( X \) and \( Y \) be topological spaces and let \( p \ \colon X \longrightarrow Y \) be a continuous map.

1. If \( p \) is a covering map, then show that \( p \) is a local homeomorphism.
2. Show that the converse of \( (1) \) is not true.
3. Show that if \( p \) is a proper local homeomorphism, then \( p \) is a covering map.