?

Free Version

Upgrade subject to access all content

Moderate

Continuity in the Co-Finite Topology On Natural Numbers

TOPO-BEY15E

Consider $\mathbb{N}$ with co-finite topology.

This means that the open sets are the empty set $\emptyset$ and all $A\subset \mathbb{N}$ such that $A^c:= \mathbb{N}\setminus A$ is a finite set. Let $a$ be the sequence $(a_n)_{n\geq 1}$ defined by $a_n:=n$.

Choose the INCORRECT statements from the list.

A

The sequence $a$ has no limit in $\mathbb{N}$,

B

The sequence $a$ has multiple limits in $\mathbb{N}$,

C

Every point $k$ in $\mathbb{N}$ is a limit of $a$,

D

Every real-valued function on $\mathbb{N}$ is a continuous map,

E

No real-valued function on $\mathbb{N}$ is a continuous map,

F

The only real-valued continuous functions on $\mathbb{N}$ are constant maps.