A map $f \colon X \to Y$ is called sequentially continuous at the point $a$ if for every sequence $(x_n)$ such that $x_n\to a$, we also have $f(x_n)\to f(a)$. $$x_n\to a \qquad \Rightarrow \qquad f(x_n) \to f(a).$$ A map $f \colon X \to Y$ is called sequentially continuous, if it is sequentially continuous at each point of $X$.
In sequential spaces sequential continuity and continuity are equivalent. In fact, this property characterizes sequential spaces in the sense that if this equivalence holds for every topological space $Y$ and for every map $X\to Y$, then $X$ is sequential. This characterization is mentioned also in Wikipedia article. Although I am not able to find a reference for this fact in a textbook or an article at this moment. (I'd say this is well-known and it can be considered folklore.)
One possible way to see this: If $X$ is not sequential and $sX$ is the sequential coreflection of $X$, then $id_X \colon X \to sX$ is sequentially continuous but not continuous. By sequential coreflection $sX$ of $X$ I mean the topological space which has the same underlying set as $X$ but closed sets are only those sets, which are sequentially closed in the original topology.
A different argument: This can also be seen from the fact that a sequential space $X$ can be obtained as a quotient space of a topological sum, where summands correspond to convergent sequences in $X$. (See, for example, proof of Theorem 3.10 in this paper.) A map from a quotient space is continuous if and only if the composition with the quotient map is continuous. Continuity of the composition map is precisely convergence of the image of corresponding sequence.
What I am interested in is the following characterization of Fréchet-Urysohn spaces (a.k.a. Fréchet spaces): A space $X$ is Fréchet-Urysohn if and only if continuity at a point is equivalent to sequential continuity at a point. (Again, in the sense that this equivalence holds for each $x\in X$, any topological space $Y$ and for each map $f\colon X\to Y$.)
This characterization was not known to me until it came up in a discussion with my colleague recently. My main question is:
- Do you happen to know about any reference for this characterization of Fréchet-Urysohn spaces?
But I'd be grateful for providing alternative proofs to the one I suggested below, too. (Although that was not the main reason for posting this question.)
I'll add sketch of the proof. I do not claim that this it the most direct proof, but for some reasons the remaining equivalent conditions were interesting for me, too.
The following conditions are equivalent for a topological space $X$:
(a) $X$ is Fréchet-Urysohn.
(b) $X$ is hereditary sequential (i.e., every subspace of $X$ is sequential).
(c) For each $a\in X$ the prime factor $X_a$ is Fréchet.
(d) For each $a\in X$ the prime factor $X_a$ is sequential.
If $a\in X$, then the prime factor $X_a$ of $X$ at $a$ is the space which has the same neighborhoods of $a$ as $X$ and all points other than $a$ are isolated. I have seen this terminology in several papers, see references in my answer here. Thomas Andrews called it local topology in one of his recent answers.
The important fact is that $f \colon X_a \to Y$ is continuous if and only $f \colon X\to Y$ is continuous at $a$.
Sketch of a proof. (a) $\Leftrightarrow$ (b): This is well-known, see [F, Proposition 7.2], [E, Exercise 2.4.6].
(a) $\Rightarrow$ (c): We want to show that if $a\in\overline V$ holds in $X_a$, then there is a sequence of points of $V$ which converges to $a$. But, since the neighborhoods of $a$ in $X_a$ and in $X$ are the same, $a\in\overline V$ holds in the space $X$, too. Since this space is Fréchet, there is a sequence of points of $V$ converging to $a$ in $X$. This sequence converges to $a$ in $X_a$, too.
(c) $\Rightarrow$ (d): Obvious.
(d) $\Rightarrow$ (b): We can use the observation that every space can be obtained as a quotient of the topological sum of all its prime factors. This shows that $X$ is sequential. If $Y$ is a subspace of $X$ and $a\in Y$ then we have a quotient map $X_a\to Y_a$ (simply by mapping points from $Y\setminus X$ to the point $a$). Using the same argument again, we get that $Y$ is sequential.
[E] R. Engelking, General Topology. Sigma Series in Pure Mathematics 6. Berlin: Heldermann, revised ed., 1989.
[F] S.P. Franklin, Spaces in which sequences suffice II, Fund. Math. 61 (1967).