# Determine all open sets in the subspace $A$ of $\mathbb{R}$.

Consider $\mathbb{R}$ with its standard topology and let $A=(0,1)\cup\{2\}$ subset of $\mathbb{R}$. Determine all open sets in the subspace $A$ of $\mathbb{R}$. This problem is in Topology by Djugundji but I cannot check my answer if it right since it does not have solution. Any help would be appreciated..

Any $X \subseteq A \cap (0,1)$ that is open in $\mathbb R$ is open in $A$ and for each of these $X \cup \{2\}$ is also open in $A$. Every open subset of $A$ has one of these two forms. I leave the easy proof of this to you (you just have to unwind the definition of the subspace topology).
• Why is $X \cup [2]$ open ? (I assume that {2} is the closed set $[2]$ - so why is the union of an open and closed set open ? – Tom Collinge Mar 11 '16 at 12:11
• @TomCollinge $\{2\}$ is open in the subspace topology, because $\{2\} = (1.5,2.5) \cap A$. – Stefan Mesken Mar 11 '16 at 12:13
• @Bayoy You can't enumerate all open sets (there are precisely $2^{\aleph_0}$ many of them). I'm pretty sure 'determine' means 'describe' in this context. – Stefan Mesken Mar 11 '16 at 12:14