Suppose a space $Y$ is given along two subspaces $Y_1,Y_2$ s.t. $Y=Y_1^{\circ}\cup Y_2^{\circ}$ and the intersection $Y_0=Y_1\cap Y_2$ is non-empty. If $(Y_1,Y_0)$ is $p$-connected and $(Y_2,Y_0)$ is $q$-connected for some $p,q\ge0$, then the inclusion induces isomorphisms $\pi_n(Y_1,Y_0,\ast)\rightarrow\pi_n(Y,Y_2,\ast)$ for $n<p+q$ and an epimorphism for $n=p+q$. This is the Blakers-Massey Theorem.
There are a variety of proofs of this Theorem, but the one I'm interested in the proof due to Puppe (with some inspiration by Boardman, apparently) that can be found in the book Homotopietheorie co-authored by tom Dieck, Kamps and Puppe himself (ch. 15) or, more recently, in tom Dieck's book Algebraic Topology (Theorem 6.9.4). Notably, however, either only proves the theorem under the stronger hypothesis that the subspaces $Y_1,Y_2$ are themselves open in $Y$. The weaker version of theorem stated above is in most other places in the literature, which makes me raise the
Question: Is it possible to adapt the Puppe proof to the more general statement of the theorem?
Following the proof in tom Dieck, the openness appears in two places. First, in the proof of what he calls the "preparation theorem" (Theorem 6.9.2). This goes through verbatim under the weaker hypotheses. The second appearance is in the proof of the intermediate Theorem 6.9.3. It comes down to obtaining a map $\psi\colon I^n\times I\rightarrow Y$ by the preparation theorem and then modifying it by a final homotopy. This homotopy is constructed using an auxiliary function $\tau\colon I^n\rightarrow I$ which is $0$ on $\pi\psi^{-1}(Y\setminus Y_1)$ and $1$ on $\pi\psi^{-1}(Y\setminus Y_2)$, where $\pi\colon I^n\times I\rightarrow I^n$ is the projection. The previous construction ensures the two sets to be disjoint, so if $Y_1,Y_2$ are open, these sets are closed and disjoint, hence $\tau$ exists. In case of the weaker hypotheses, we would instead have to ensure that, say, $\overline{\psi^{-1}(Y\setminus Y_1)}$ and $\overline{\psi^{-1}(Y\setminus Y_2)}$ have disjoint images under $\pi$ and I have not been able to modify the constructions to obtain this.