Closed graph theorem on topological space requires the image space Y is compact Hausdorff. Some counter examples on Wikipedia are all non-Hausdorff which is considered more exotic examples.
What are the examples in which the map X->Y is continuous but not closed (as defined in http://en.wikipedia.org/wiki/Closed_graph_theorem, not the closed map in general topology), and the image space Y is Hausdorff but not compact?
Best Answer
There is no such example. Aa long as $Y$ is Hausdorff the graph of $f$ is closed : Let $(x_i,f(x_i))_{i \in I}$ be a net in the graph converging to $(x,y)$ in the product topology. Then $x_i \to x$ and $f(x_i) \to y$. By continuity $f(x_i) \to f(x)$. Since limits are unique in a Hausdorff space we get $y=f(x)$. Hence the graph is closed.