# topological properties and nets

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

## 1 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 (http://planetmath.org/Net) to $x$.

• For a detailed proof, click here (http://planetmath.org/NetsAndClosuresOfSubspaces).

## 2 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 $Y$.

## 3 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.

## 4 Hausdorff

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

## 5 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 (http://planetmath.org/CompactnessAndConvergentSubnets).

## 6 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 (http://planetmath.org/ContinuityAndConvergentNets).

## 7 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 to a net $\{x_{i_{j}}\}_{j\in J}\subset X$ such that $x_{i_{j}}\longrightarrow x$. By ”” we that $\{x_{i_{j}}\}_{j\in J}$ is such that $f(x_{i_{j}})=y_{i_{j}}$.

• For a detailed proof, click here (http://planetmath.org/ContinuousSurjectiveOpenMapsInTermsOfNets).

## 8 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 $f_{\alpha}(x)$.

### 8.0.1 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 $X$.

### 8.0.2 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 $x^{\alpha}$.

## 9 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 $f(x)$.

Title topological properties and nets TopologicalPropertiesAndNets 2013-03-22 18:38:04 2013-03-22 18:38:04 asteroid (17536) asteroid (17536) 6 asteroid (17536) Feature msc 54A20