Non-Hausdorff manifolds via adjunction spaces (Q2683746)

Given collections \(\mathcal T=\{X_i\mid i\in I\}\) of topological spaces, \(\mathcal A=\{A_{ij}\mid i,j\in I\}\) with \(A_{ij}\subset X_i\) for each \(j\), and \(\mathcal E=\{e_{ij}:A_{ij}\to X_j\mid i,j\in I\}\) of embeddings such that \(A_{ii}=X_i\), \(e_{ii}=\mathrm{id}\), \(A_{ji}=e_{ij}(A_{ij})\), \(e_{ij}^{-1}=e_{ji}\) and \(e_{ik}=e_{jk}e_{ij}\), the adjunction space \(\bigcup_{(\mathcal T,\mathcal A,\mathcal E)}X_i\) is the quotient space of \(\bigsqcup_iX_i=\{(x,i)\ /\ x\in X_i\}\) under the equivalence relation \((x,i)\sim(y,j)\) when \(e_{ij}(x)=y\). Particular attention is paid to the case where \(I\) is countable, each \(X_i\) is a second countable (Hausdorff) topological manifold and each \(A_{ij}\) is open, with some properties deduced. In particular, it is shown that all second countable, locally euclidean spaces may be obtained in this way. Characterisations of open, Hausdorff, connected subspaces that are maximal are explored. A number of relevant examples are presented.