Show that the diagonal map maps X homeomorphically onto the diagonal

Question

Let $X$ be a topological space and let $ ∆ = \{(x; x) : x ∈ X\}$ be the diagonal of $X$.

Define $f: X → X × X$ by $f(x) = (x; x)$. Show that $f$ maps $X$ homeomorphically onto $ ∆$.

Answer 1

Showing that map to be bijective is trivial. Note that any $\Delta$-open set can without loss of generality be written as the union of sets of the form $\Delta\cap(U\times U)$, where $U$ is $X$-open. This follows from the definition of the product topology on $X\times X$, and from the topology it induces on $\Delta$ as a subspace (plus a bit of finagling). This will make it easier to show continuity. Also, the image of any $X$-open set $U$ under the map is equal to $\Delta\cap(U \times U)$, which can be shown without much difficulty, and from that it follows that the map is open.