Difficult# Converging Nets

TOPO-NEDXK5

Which of the following nets converge(s)? Select **ALL** that apply.

A

The sequence $a:\mathbb N\rightarrow\mathbb Z$ sending $n\mapsto (-1)^n$ denoted by $a_n$, where $\mathbb N$ as a directed set is ordered according to the standard ordering of the reals.

B

The function $f:\mathbb R\rightarrow\mathbb R$ sending $x\mapsto e^x$ where $\mathbb R$ as a directed set is ordered in reverse to the standard ordering of the reals.

C

The probability measure $P:\mathcal F\mapsto [0,1]$ of a probability space $(\Omega,\mathcal F,P)$, sending events to their probabilities, where $\mathcal F$ as a directed set is partially ordered by inclusion.

D

A counting function $c:\mathcal F\rightarrow\mathbb Z_{\ge 0}$ sending $S\mapsto|S|$, where $\mathcal F$ is the set of all finite subsets of $\mathbb N$, and $\mathcal F$ as a directed set is partially ordered by inclusion.

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