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 $x$.

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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.

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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 $f(x)$.

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For a detailed proof, click here (http://planetmath.org/ContinuityAndConvergentNets).
7 Open map
Let $f:X\u27f6Y$ 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 ${y}_{i}\u27f6y$, 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 ${x}_{{i}_{j}}\u27f6x$. By ”” we that ${\{{x}_{{i}_{j}}\}}_{j\in J}$ is such that $f({x}_{{i}_{j}})={y}_{{i}_{j}}$.

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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 Compactopen 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 compactopen 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 

Canonical name  TopologicalPropertiesAndNets 
Date of creation  20130322 18:38:04 
Last modified on  20130322 18:38:04 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  6 
Author  asteroid (17536) 
Entry type  Feature 
Classification  msc 54A20 