Topology

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Moderate

Maps and Convergence of Nets

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Let $(X,\mathcal{T})$ be a topological space, $A$ a directed set, and $a:A\to X$ a net in $X$. Assume in addition that for every point $x\in X$ and every open neighborhood $U$ of $x$, one can find a function $f_{x,U}:X\to \mathbb{R}$, continuous at $x$, and satisfying $f_{x,U}(x)=1$, and $f_{x,U}(y)=0$, for all $ y\not\in U_x$.

Find the four mutually equivalent statements in the following answer choices.

A

$a$ is a convergent net, and its limit is $x_a\in X$.

B

If $y\in X$ is such that for every open neighborhood $U$ of $y$, one can find $\beta\in A$ such that $a_\alpha\in U$ for all $\alpha\in A$ satisfying $\,\alpha \geq \beta$ , then $y=x_a$.

C

For every $f:X\to Y$ a function to another topological space $Y$, we have that $f( a):A \to Y$ is a convergent net in $Y$.

D

For every $f:X\to Y$ a continuous function at $x_a$, we have that $f(a):A \to Y$ is a convergent net in $Y$.

E

For every open neighborhood $U$ of $x_a$, one can find $\beta\in A$ such that $a_\alpha\in U$ for all $\alpha\in A$ satisfying $\,\alpha \geq \beta$.

F

For every $f:X\to Y$ a continuous function on $X$, we have $f(a):A \to Y$ is a convergent net in $Y$, and its limit is $f(x_a)$.