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Topology

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Nets and Sequences: a Basic Comparison

TOPO-VWGEJC

Let $X$ be a topological space.

Choose the correct statements from the list. Select ALL that apply.

A

Every net $n:A\to X$ is a sequence.

B

Every sequence $a:\mathbb{N}\to X$ is a net.

C

If $A$ is an infinitely countable directed set, then a net $n:A\to X$ can be matched to a sequence $x:\mathbb{N}\to X$ in the following way: there exists a bijection $f:A\to \mathbb{N},$ such that $n(a)=x_{f(a)}$, for each $a\in A$.

D

If $A$ is an infinitely countable directed set, then a net $n:A\to X$ can be matched to a sequence $x:\mathbb{N}\to X$ in the following way: the bijection $f$ from part C satisfies in addition that $a\leq b$ in $A$ if and only if $f(a)\leq f(b)$ in $\mathbb{N}$.