Constant Net is Convergent

Theorem

Let $\struct {X, \tau}$ be a topological space.

Let $\struct {\Lambda, \preceq}$ be a directed set.

Let $x \in X$.

Define a net $\family {x_\lambda}_{\lambda \in \Lambda}$ by:

$x_\lambda = x$ for each $\lambda \in \Lambda$.

Then $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$.

Proof

Let $U$ be an open neighborhood of $x$.

Let $\lambda_0 \in \Lambda$.

Then we have $x_\lambda \in U$ for all $\lambda \in \Lambda$, and in particular all $\lambda \in \Lambda$ with $\lambda_0 \preceq \lambda$.

So $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$.

$\blacksquare$