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Topology

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Moderate

Convergent Nets and Operations

TOPO-KIYLDN

Let $(X,\mathcal{T})$ be a topological algebra, which means that addition and multiplication, as well as multiplication by scalars, are defined on $X$, and all of these operations are continuous with respect to $\mathcal{T}$. Let $A$ be a directed set. Assume that $a:A\to X$, $b:A\to X$ and $c:A\to X$ are all nets in $X$.

If $a$ and $b$ are convergent nets in $X$, which of the following statements are necessarily true? Select ALL that apply.

A

$3 a+ 2 b$ is a convergent net in $X$.

B

There exists $x_a\in X$ such that for any open neighborhood $U$ of $x_a$ there exists $\beta\equiv \beta(U)$ such that $a_\alpha\in U$ for all $\alpha\in A$ whenever $\alpha \geq \beta$.

C

$b\cdot c$ is a convergent net in $X$.

D

If $P_m$ is a polynomial of degree $m$, then $(P_m(a_\alpha))_{\alpha\in A}$ converges to $P_m (x_a)$, where $x_a$ is the limit of $a$.

E

$2a-\pi b\cdot c$ is a convergent net.

F

$b\cdot a$ is a convergent net in $X$.