Is there a topological space $X$ such that $C_p(X)$ is sequential but not Fréchet?


  1. A topological space $X$ is called sequential if whenever $A \subset X$ contains all limits of sequences of points from itself, then $A$ is closed in $X$.

  2. A topological space $X$ is called Fréchet if whenever $x \in \overline A$, there exists a sequence in $A$ which converges to $x$.

  3. $C_p(X)$ is the space of continuous functions from $X$ to $\mathbb R$ with the topology of pointwise convergence. That is, the topology generated by the base consisting of sets of the form $$[x_1,...,x_n,(a,b)] = \{ f : X \rightarrow \mathbb R \mid f(x_1),f(x_2),...,f(x_n) \in (a,b) \}$$ for $x_1 , \ldots , x_n \in X$ and real numbers $a < b$.

For more detailed discussion of these terms one can see S. P. Franklin's Spaces in which sequences suffice, (Fund. Math. vol.57 (1965) pp.107–115, link).

I do have a topological space which is sequential but not Fréchet. But, as far as I know, it is not $C_p(X)$ for any space $X$


No such space exists.

It was proved by J. Gerlits in Some properties of $C(X)$, II (Top. Appl., vol.15, No.3, pp.255-263, link, doi: 10.1016/0166-8641(83)90056-1) that for a Tychonoff space (T3 1/2-space) $X$ the following conditions are equivalent:

  1. $C_p(X)$ is Fréchet-Urysohn;
  2. $C_p(X)$ is sequential;
  3. $C_p(X)$ is a k-space (meaning that given any non-closed $A \subseteq C_p (X)$ there is a compact $K \subseteq C_p(X)$ such that $A \cap K$ is not closed); and
  4. $X$ has property (γ) (which means that whenever $\mathcal{U}$ is an open cover of $X$ with the property that given any $x_1 , \ldots , x_n \in X$ there is a $U \in \mathcal{U}$ containing all of then (i.e., $\mathcal{U}$ is an ω-cover), then there is a sequence $\langle U_n \rangle_{n \in \omega}$ in $\mathcal{U}$ such that $\bigcup_n \bigcap_{n \leq k} U_n = X$).

This solves the problem because for any space $X$ there is a Tychonoff space $Y$ such that $C_p(X)$ and $C_p(Y)$ are "topologically isomorphic": there is a homeomorphism $\varphi : C_p(X) \to C_p(Y)$ which preserves sums and products (i.e., $\varphi ( f + g ) = \varphi ( f ) + \varphi (g)$ and $\varphi ( f \cdot g ) = \varphi ( f ) \cdot \varphi ( g )$). (For this see V. V. Tkachuk's A $C_p$-Theory Problem Book, problem 100, p.11, and its solution pp.90-92.)


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