Let $(X, \tau)$ be a topological space. The following are equivalent:
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1. $(X, \tau)$ is Hausdorff.
2. The **diagonal** $\Delta = \{(x, x) : x \in X\}$ is closed in the [product topology](/page/Product%20Topology) on $X \times X$.
3. For every pair of continuous maps $f, g: Y \to X$ from a topological space $Y$, the **equaliser** $\{y \in Y : f(y) = g(y)\}$ is closed in $Y$.