Is $S^1$ an open subspace of a contractible space?

Question

Is $S^1$ an open subspace of a contractible space?

Answer 1

The space $\mathbb{C}/\mathbb{R}_{>0}$ is obtained by adding a point to $S^1$ in such a way that the added point is closed and the topology is the coarsest possible under these conditions (so that a subset is open iff it is an open subset of $S^1$ or the whole space).

The subspace $S^1=\mathbb{C}^\times/\mathbb{R}_{>0}$ of $\mathbb{C}/\mathbb{R}_{>0}$ is open, and the ambient space is contractible (by passing to the quotient the retracting homotopy $(z,t)\mapsto tz$).

As erz said in the comments, this is not possible if you want your contractible space to be Hausdorff, because otherwise $S^1$ would be one of its connected components.

Answer 2

Let $K=\{\emptyset, \{p\}, \{p, q\}\}$ be the Kuratowski pair. The quotient of the product space $S^1\times K$ by $S^1\times \{q\}$ is contractible and contains $S^1$ as an open subspace.

This is imitating the cone construction of contractible non-locally contractible spaces.