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A topological space $X$ is sequential if it is true that an any map whose domain is $X$ is continuous iff it is sequentially continuous.

For a projective countable spectrum of sequential topological spaces (i.e. the sequence is connected with a system of continuous maps satisfying a commuatative-diagramm condition) the limit may fail to be sequential even if all steps $X_{n}$ are Fr\'{e}chet-Uryson-spaces (see Engelking/ General Topology).

But how is the situation for projective sequences of sequential LCVS (with the embeddings as connecting maps), i.e. for a sequence $(E_{n})_{n \in \mathbb N }$ such that $E_{n+1}$ embedds continuously into $E_{n}$? Clearly, if a LCVS is sequential, then it must be bornological. Bornologicity of $\varprojlim X_{n}$ is for some situation equivalent to the vanishing of $Proj^{1}(X_{n})$ (see for example the book by Wengenroth).

Is there any general theorem which guarantees that the projective limit is sequential if the all the steps are so?

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Since your working with embeddings, in this case the projective limits is simply the intersection $\bigcap\limits_{n\in\mathbb N} E_n$, right? –  Martin Sleziak Aug 5 '12 at 16:12
    
Yes, that's right. –  Sebastian Aug 6 '12 at 8:05

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