Question

Show that if X is Hausdorff, a net in X converges to atmost one point.

Answer 1

Suppose X is a topological space where each convergent net has a unique limit. If X were not Hausdorff, then there would exist a pair of points x and y such that for any open sets $V\in\mathcal{V}_x$ and $U\in\mathcal{V}_y$ there is $x_{U,V}\in V\cap U$ . then $\{x_{V,U}:(V,U)\in\mathcal{V}_x\times\mathcal{V}_y\}$ is a net in X that converges to both x

x and y which is a contradiction. Conversely, suppose X

X is Hausdorff and {{xn:n∈D} is net converging to x

x and y. If x≠y, let Vx and Vy be disjoint open neighborhoods of x

x and y respectively. There is m∈D such that $x_n\in V_x$  and $x_n\in V_y$  for all n≥m.This is a contradictio to $V_x\cap V_y=\emptyset$