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The second question has negative answer: just consider the unit interval $[0,1]$ and let $\mathcal S$ be the family of all closed subsets with a unique non-isolated point in $[0,1]$. The family $\mathcal S$ is endowed with the discrete topology. Let $X=\{(x,S):x\in S\in\mathcal S\}\subset [0,1]\times\mathcal S$ be the topological sum of the family $\mathcal S$ and $q:X\to[0,1]$, $q:(x,S)\mapsto x$, be the natural projection. It is easy to see that the map $q$ is quotient but $q(K)\ne [0,1]$ for any compact subset $K\subset X$. So, the space $X$ is the topological sum of all convergent sequences in $[0,1]$. It is a locally compact locally countable space of density continuum.

It seems that the (metrizable locally compact locally countable) space $X$ and the equivalence relation $\sim=\{(x,y)\in X\times X:q(x)=q(y)\}$ yield also a counterexample to the first question for $Y=\mathbb R$ or $Y=\{0,1\}$.

The second question has negative answer: just consider the unit interval $[0,1]$ and let $\mathcal S$ be the family of all closed subsets with a unique non-isolated point in $[0,1]$. The family $\mathcal S$ is endowed with the discrete topology. Let $X=\{(x,S):x\in S\in\mathcal S\}\subset [0,1]\times\mathcal S$ be the topological sum of the family $\mathcal S$ and $q:X\to[0,1]$, $q:(x,S)\mapsto x$, be the natural projection. It is easy to see that the map $q$ is quotient but $q(K)\ne [0,1]$ for any compact subset $K\subset X$. So, the space $X$ is the topological sum of all convergent sequences in $[0,1]$. It is a locally compact locally countable space of density continuum.

It seems that the (metrizable locally compact locally countable) space $X$ and the equivalence relation $\sim=\{(x,y)\in X\times X:q(x)=q(y)\}$ yield also a counterexample to the first question for $Y=\mathbb R$.

The second question has negative answer: just consider the unit interval $[0,1]$ and let $\mathcal S$ be the family of all closed subsets with a unique non-isolated point in $[0,1]$. The family $\mathcal S$ is endowed with the discrete topology. Let $X=\{(x,S):x\in S\in\mathcal S\}\subset [0,1]\times\mathcal S$ be the topological sum of the family $\mathcal S$ and $q:X\to[0,1]$, $q:(x,S)\mapsto x$, be the natural projection. It is easy to see that the map $q$ is quotient but $q(K)\ne [0,1]$ for any compact subset $K\subset X$. So, the space $X$ is the topological sum of all convergent sequences in $[0,1]$. It is a locally compact locally countable space of density continuum.

It seems that the (metrizable locally compact locally countable) space $X$ and the equivalence relation $\sim=\{(x,y)\in X\times X:q(x)=q(y)\}$ yield also a counterexampel to the first question for $Y=\mathbb R$ or $Y=\{0,1\}$.

The second question has negative answer: just consider the unit interval $[0,1]$ and let $\mathcal S$ be the family of all closed subsets with a unique non-isolated point in $[0,1]$. The family $\mathcal S$ is endowed with the discrete topology. Let $X=\{(x,S):x\in S\in\mathcal S\}\subset [0,1]\times\mathcal S$ be the topological sum of the family $\mathcal S$ and $q:X\to[0,1]$, $q:(x,S)\mapsto x$, be the natural projection. It is easy to see that the map $q$ is quotient but $q(K)\ne [0,1]$ for any compact subset $K\subset X$. So, the space $X$ is the topological sum of all convergent sequences in $[0,1]$. It is a locally compact locally countable space of density continuum.

It seems that the (metrizable locally compact locally countable) space $X$ and the equivalence relation $\sim=\{(x,y)\in X\times X:q(x)=q(y)\}$ yield also a counterexample to the first question for $Y=\mathbb R$ or $Y=\{0,1\}$.

The second question has negative answer: just consider the unit interval $[0,1]$ and let $\mathcal S$ be the family of all closed subsets with a unique non-isolated point in $[0,1]$. The family $\mathcal S$ is endowed with the discrete topology. Let $X=\{(x,S):x\in S\in\mathcal S\}\subset [0,1]\times\mathcal S$ be the topological sum of the family $\mathcal S$ and $q:X\to[0,1]$, $q:(x,S)\mapsto x$, be the natural projection. It is easy to see that the map $q$ is quotient but $q(K)\ne [0,1]$ for any compact subset $K\subset X$. So, the space $X$ is the topological sum of all convergent sequences in $[0,1]$. It is a locally compact locally countable spaces of density continuum.

It seems that the (metrizable locally compact locally countable) space $X$ and the equivalence relation $\sim=\{(x,y)\in X\times X:q(x)=q(y)\}$ yield also a counterexampel to the first question for $Y=\mathbb R$ or $Y=\{0,1\}$.

The second question has negative answer: just consider the unit interval $[0,1]$ and let $\mathcal S$ be the family of all closed subsets with a unique non-isolated point in $[0,1]$. The family $\mathcal S$ is endowed with the discrete topology. Let $X=\{(x,S):x\in S\in\mathcal S\}\subset [0,1]\times\mathcal S$ be the topological sum of the family $\mathcal S$ and $q:X\to[0,1]$, $q:(x,S)\mapsto x$, be the natural projection. It is easy to see that the map $q$ is quotient but $q(K)\ne [0,1]$ for any compact subset $K\subset X$. So, the space $X$ is the topological sum of all convergent sequences in $[0,1]$. It is a locally compact locally countable space of density continuum.

It seems that the (metrizable locally compact locally countable) space $X$ and the equivalence relation $\sim=\{(x,y)\in X\times X:q(x)=q(y)\}$ yield also a counterexampel to the first question for $Y=\mathbb R$ or $Y=\{0,1\}$.