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Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ belongs to the first Baire class (to the first homotopic Baire class), if there exists a continuous map $H:X\times \omega\to Y$ (a continuous map $H:X\times [0,+\infty)\to Y$) such that \begin{gather} \lim_{n\to\infty} H(x,n)=f(x) \end{gather} for every $x\in X$. The collection of all (homotopic) Baire-one maps between $X$ and $Y$ we denote by ${\rm B}_1(X,Y)$ (${\rm hB}_1(X,Y)$).

Evidently, ${\rm hB}_1(X,Y)\subseteq {\rm B}_1(X,Y)$ for any spaces $X$, $Y$, and ${\rm hB}_1(X,Y)={\rm B}_1(X,Y)$ if either $X$ or $Y$ is contractible. Moreover, ${\rm hB}_1(X,Y)={\rm B}_1(X,Y)$ if $X$ is a topological space and $Y$ is a metrizable ANR.

Do there exist path-connected subspaces $X,Y\subseteq\mathbb R^2$ such that ${\rm hB}_1(X,Y)\ne {\rm B}_1(X,Y)$?

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  • $\begingroup$ Does $hB_1$ mean considering the maps 'up to homotopy'? In that case why is $hB_1 \subset B_1$? $\endgroup$ – Daron Mar 14 '18 at 9:37
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    $\begingroup$ $f\in B_1(X,Y)$ ($f\in hB_1(X,Y)$) if there exists a sequence of continuous maps $f_n:X\to Y$ such that ($f_n$ is homotopic to $f_k$ for all $n,k$ and) $(f_n)$ converges to $f$ pointwisely on $X$. $\endgroup$ – MasleniZZa Mar 14 '18 at 10:05

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