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First, there can't be a path from $c_0$ to $c_1$, else a continuous map would give a path from $x$ to $y$. By the same argument, $C$ is not allowed to have finitely many or countably many path components. If one can build something like the topologist's sine curve on a very long line of arbitrarily large cardinality, the number of path components of $C$ has to be larger or equal than any cardinal number. There is no largest cardinal number. Hence, the set of points of $C$ is no set.

Thus: If there exist long lines in every cardinality, the answer to your question is no.

Edit: OK, if $x$ and $y$ are uncountably far apart, there is no path from $x$ to $y$. Call the set of points between $x$ and $y$ the "long path" from $x$ to $y$. It contains an arbitrarily large number of connected non-path-connected, pairwise disjoint sub"paths". Say the number is $\kappa '$. The connected components of their preimages under $f$ also have to be non-path-connected and pairwise disjoint. Hence, the universal space $C$ that we are looking for needs to have at least $\kappa '$ path components, hence at least $\kappa '$ points. If the "very long line" exists for every cardinality, the universal space $(C, c_0,c_1)$ can't exist.

Edit: Goldstern's argument is much simpler than this. Just for completeness: Very long lines exist (see Goldstern's comment) and so the answer is definitely No.

First, there can't be a path from $c_0$ to $c_1$, else a continuous map would give a path from $x$ to $y$. By the same argument, $C$ is not allowed to have finitely many or countably many path components. If one can build something like the topologist's sine curve on a very long line of arbitrarily large cardinality, the number of path components of $C$ has to be larger or equal than any cardinal number. There is no largest cardinal number. Hence, the set of points of $C$ is no set.

Thus: If there exist long lines in every cardinality, the answer to your question is no.

Edit: OK, if $x$ and $y$ are uncountably far apart, there is no path from $x$ to $y$. Call the set of points between $x$ and $y$ the "long path" from $x$ to $y$. It contains an arbitrarily large number of connected non-path-connected, pairwise disjoint sub"paths". Say the number is $\kappa '$. The connected components of their preimages under $f$ also have to be non-path-connected and pairwise disjoint. Hence, the universal space $C$ that we are looking for needs to have at least $\kappa '$ path components, hence at least $\kappa '$ points. If the "very long line" exists for every cardinality, the universal space $(C, c_0,c_1)$ can't exist.

Edit: Goldstern's argument is much simpler than this. Just for completeness: Very long lines exist (see Goldstern's comment) and so the answer is definitely No.

Edit: The concept of homotopy theory with respect to a big interval is actually worked with. I just stumbled over a preprint by Penrod in the field and the foundations seem to be laid by Cannon-Gonner [J. Cannon and G. Conner, The big fundamental group, big Hawaiian earrings, and the big free groups, Topology Appl. 106 (2000)].

First, there can't be a path from $c_0$ to $c_1$, else a continuous map would give a path from $x$ to $y$. By the same argument, $C$ is not allowed to have finitely many or countably many path components. If one can build something like the topologist's sine curve on a very long line of arbitrarily large cardinality, the number of path components of $C$ has to be larger or equal than any cardinal number. There is no largest cardinal number. Hence, the set of points of $C$ is no set.

Thus: If there exist long lines in every cardinality, the answer to your question is no.

Edit: OK, if $x$ and $y$ are uncountably far apart, there is no path from $x$ to $y$. Call the set of points between $x$ and $y$ the "long path" from $x$ to $y$. It contains an arbitrarily large number of connected non-path-connected, pairwise disjoint sub"paths". Say the number is $\kappa '$. The connected components of their preimages under $f$ also have to be non-path-connected and pairwise disjoint. Hence, the universal space $C$ that we are looking for needs to have at least $\kappa '$ path components, hence at least $\kappa '$ points. If the "very long line" exists for every cardinality, the universal space $(C, c_0,c_1)$ can't exist.

First, there can't be a path from $c_0$ to $c_1$, else a continuous map would give a path from $x$ to $y$. By the same argument, $C$ is not allowed to have finitely many or countably many path components. If one can build something like the topologist's sine curve on a very long line of arbitrarily large cardinality, the number of path components of $C$ has to be larger or equal than any cardinal number. There is no largest cardinal number. Hence, the set of points of $C$ is no set.

Thus: If there exist long lines in every cardinality, the answer to your question is no.

Edit: OK, if $x$ and $y$ are uncountably far apart, there is no path from $x$ to $y$. Call the set of points between $x$ and $y$ the "long path" from $x$ to $y$. It contains an arbitrarily large number of connected non-path-connected, pairwise disjoint sub"paths". Say the number is $\kappa '$. The connected components of their preimages under $f$ also have to be non-path-connected and pairwise disjoint. Hence, the universal space $C$ that we are looking for needs to have at least $\kappa '$ path components, hence at least $\kappa '$ points. If the "very long line" exists for every cardinality, the universal space $(C, c_0,c_1)$ can't exist.

Edit: Goldstern's argument is much simpler than this. Just for completeness: Very long lines exist (see Goldstern's comment) and so the answer is definitely No.

First, there can't be a path from $c_0$ to $c_1$, else a continuous map would give a path from $x$ to $y$. By the same argument, $C$ is not allowed to have finitely many or countably many path components. If one can build something like the topologist's sine curve on a very long line of arbitrarily large cardinality, the number of path components of $C$ has to be larger or equal than any cardinal number. There is no largest cardinal number. Hence, the set of points of $C$ is no set.

Thus: If there exist long lines in every cardinality, the answer to your question is no.

Edit: OK, if $x$ and $y$ are uncountably far apart, there is no path from $x$ to $y$. Sorry. The argument could still work with "paths" that are uncountably long. But now I get tangled up in set theory. Maybe this approach does not give a counterexample after all.

First, there can't be a path from $c_0$ to $c_1$, else a continuous map would give a path from $x$ to $y$. By the same argument, $C$ is not allowed to have finitely many or countably many path components. If one can build something like the topologist's sine curve on a very long line of arbitrarily large cardinality, the number of path components of $C$ has to be larger or equal than any cardinal number. There is no largest cardinal number. Hence, the set of points of $C$ is no set.

Thus: If there exist long lines in every cardinality, the answer to your question is no.

Edit: OK, if $x$ and $y$ are uncountably far apart, there is no path from $x$ to $y$. Call the set of points between $x$ and $y$ the "long path" from $x$ to $y$. It contains an arbitrarily large number of connected non-path-connected, pairwise disjoint sub"paths". Say the number is $\kappa '$. The connected components of their preimages under $f$ also have to be non-path-connected and pairwise disjoint. Hence, the universal space $C$ that we are looking for needs to have at least $\kappa '$ path components, hence at least $\kappa '$ points. If the "very long line" exists for every cardinality, the universal space $(C, c_0,c_1)$ can't exist.