topological properties and nets

locally compact Hausdorff space

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 $f:X⟶Y$ 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 $y_i⟶y$, 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 $x_{i_j}⟶x$. 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, ${\{X_{\alpha }\}}_{\alpha \in 𝒜}$ 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 $\alpha \in 𝒜$, ${(f_{\alpha }(x_i))}_{i\in I}$ 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 ${\{X_{\alpha }\}}_{\alpha \in 𝒜}$ 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 $f(x)$.