Many topological properties and concepts can be translated in terms of convergence of nets. In this entry we give a list of this correspondence of properties. For detailed proofs please follow the links.

Closure

Let $X$ be a topological space and $Y \subseteq X$ a subset. A point $x \in X$ is in the closure of $Y$ if and only there exists a net in $Y$ converging to $x$ .

• For a detailed proof, click here.

Closed

Let $X$ be a topological space. A subset $Y \subseteq X$ is closed if and only if every convergent net in $Y$ converges to a point in $Y$ .

Limit point

Let $X$ be a topological space and $Y \subseteq X$ a subset. A point $x \in X$ is a limit point of $Y$ if and only if there is a net in $Y$ converging to $x$ that is not eventually constant.

Hausdorff

A topological space $X$ is Hausdorff if and only if every convergent net in $X$ has a unique limit.

Compact

A topological space $X$ is compact if and only if every net in $X$ has a convergent subnet.

• For a detailed proof, click here.

Continuous

Let $X$ and $Y$ be topological spaces. A function $f:X \to Y$ is continuous at a point $x \in X$ if and only if for every net $(x_i)_{i \in I}$ in $X$ converging to $x$ , the net $(f(x_i))_{i \in I}$ converges to $f(x)$ .

• For a detailed proof, click here.

Open map

Let $f:X \longrightarrow Y$ be a surjective map between the topological spaces $X$ and $Y$ . Then $f$ is an open mapping if and only if given a net $\{y_i\}_{i \in I} \subset Y$ such that $y_i \longrightarrow y$ , then for every $x \in f^{-1}(\{y\})$ there exists a subnet $\{y_{i_j}\}_{j \in J}$ that lifts to a net $\{x_{i_j}\}_{j \in J} \subset X$ such that $x_{i_j} \longrightarrow x$ . By "lift" we mean that $\{x_{i_j}\}_{j \in J}$ is such that $ f(x_{i_j}) = y_{i_j}$ .

• For a detailed proof, click here.

Initial topology

Let $X$ be a set, $\{X_{\alpha}\}_{\alpha \in \mathcal{A}}$ a family of topological spaces and $f_{\alpha}: X \to X_{\alpha}$ a family of functions.

A net $(x_i)_{i \in I}$ in $X$ converges to a point $x$ in the initial topology of $X$ (relatively to the mappings $f_{\alpha}$ ) if and only if for each $\alpha \in \mathcal{A}$ , $(f_{\alpha}(x_i))_{i \in I}$ converges to $f_{\alpha}(x)$ .

Particular case: subspace topology

Let $X$ be a toplogical space and $Y \subseteq X$ a subset. A net $(y_i)_{i \in I}$ in $Y$ converges to $y \in Y$ in the subspace topology if and only if $(y_i)_{i \in I}$ converges to $y$ in the topology of $X$ .

Particular case: product topology

Let $\{X_{\alpha}\}_{\alpha \in \mathcal{A}}$ be a collection of topological spaces and $X := \prod_{\alpha} X_{\alpha}$ their Cartesian product. A net $(x_i)_{i \in I}$ in $X$ converges to $x$ in the product topology if and only if every coordinate $(x_i^{\alpha})_{i \in I}$ converges to $x^{\alpha}$ .

Compact-open topology

Let $X$ be a locally compact Hausdorff space, $Y$ a topological space and $C(X, Y)$ the set of continuous functions from $X$ to $Y$ . A net $(f_i)_{i \in I}$ in $C(X, Y)$ converges to $f$ in the compact-open topology if and only if whenever a net $(x_i)_{i \in I}$ in $X$ , indexed by the same directed set $I$ , converges to $x \in X$ , we also have that $(f_i(x_i))_{i \in I}$ converges to $f(x)$ .