I've been question myself if the subspace of the real where we map a topological space needs to be necessarily open since the function that do this map is bijective and continuous, someone can help me please ?
2026-03-27 11:49:12.1774612152
A homeomorphism essentially takes a topological space into open subspace of the real?
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“Silly” counterexample: $[0,1]$ embeds into $\Bbb R$ through homeomorphism to the non-open $[0,1]$.
Less trivial example: every finite dimensional real normed vector space, say of dimension $n$, embeds (with the norm topology) into $\Bbb R^k$ for $k\gt n$ as a proper closed and nonopen subset.
So, the answer is no. Continuity does not imply openness of the image I’m afraid. Knowing when an image is open is in general a difficult task, and theorems related to this (e.g. the open mapping theorem for Banach spaces) are quite important (since it is in general false).