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There is no such continuum. See

Z. Waraszkiewicz, Sur un problème de M.H. Hahn, Fund. Math. 22 (1934) 180–205.

Waraszkiewicz constructed an uncountable family $W$ of continua in the plane called Waraszkiewicz spirals so that no continuum can be mapped continuously onto every continuum in $W$. Start with the space $X=S^1\cup [1,2]$ and consider the maps $f,g:X\to X$ where

• $f(x)=x$ if $x\in S^1$, $f(x)=\exp(4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $f(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$
• $g(x)=x$ if $x\in S^1$, $g(x)=\exp(-4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $g(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$

Now $W$ is the collection of all inverse limits of all inverse sequences $(X_i,h_i)$ where $X_i=X$ for all $i$ and $h_i\in\{f,g\}$.

For a short and readable description (where I found the construction) see:

W.T. Ingram, Concerning images of continua, Topology Proceedings 16 (1991) 89-93.

This non-existence results has been improved upon a few times, for instance in the following:

S.B. Nadler, The nonexistence of almost continuous surjections between certain continua, Topology and its Applications, 154 (2007) 1008-1014.

There is no such continuum. See

Z. Waraszkiewicz, Sur un problème de M.H. Hahn, Fund. Math. 22 (1934) 180–205.

Waraszkiewicz constructed an uncountable family $W$ of continua in the plane called Waraszkiewicz spirals so that no continuum can be mapped continuously onto every continuum in $W$. Start with the space $X=S^1\cup [1,2]$ and consider the maps $f,g:X\to X$ where

• $f(x)=x$ if $x\in S^1$, $f(x)=\exp(4\pi ix)$ for $1\leq x\leq \frac{3}{2}$, and $f(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$
• $g(x)=x$ if $x\in S^1$, $g(x)=\exp(-4\pi ix)$ for $1\leq x\leq \frac{3}{2}$, and $g(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$

Now $W$ is the collection of all inverse limits of all inverse sequences $(X_i,h_i)$ where $X_i=X$ for all $i$ and $h_i\in\{f,g\}$.

For a short and readable description (where I found the construction) see:

W.T. Ingram, Concerning images of continua, Topology Proceedings 16 (1991) 89-93.

This non-existence results has been improved upon a few times, for instance in the following:

S.B. Nadler, The nonexistence of almost continuous surjections between certain continua, Topology and its Applications, 154 (2007) 1008-1014.

There is no such continuum. See

Z. Waraszkiewicz, Sur un problème de M.H. Hahn, Fund. Math. 22 (1934) 180–205.

Waraszkiewicz constructed an uncountable family $W$ of continua in the plane called Waraszkiewicz spirals so that no continuum can be mapped continuously onto every continuum in $W$. Start with the space $X=S^1\cup [1,2]$ and consider the maps $f,g:X\to X$ where

• $f(x)=x$ if $x\in S^1$, $f(x)=\exp(4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $f(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$
• $g(x)=x$ if $x\in S^1$, $g(x)=\exp(-4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $g(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$

Now $W$ is the collection of all inverse limits of all inverse sequences $(X_i,h_i)$ where $X_i=X$ for all $i$ and $h_i\in\{f,g\}$.

For a short and readable description (where I found the construction) see:

W.T. Ingram, Concerning images of continua, Topology Proceedings 16 (1991) 89-93.

This non-existence results has been improved upon a few times, for instance in the following:

S.B. Nadler, The nonexistence of almost continuous surjections between certain continua, Topology and its Applications, 154 (2007) 1008-1014.

There is no such continuum. See

Z. Waraszkiewicz, Sur un problème de M.H. Hahn, Fund. Math. 22 (1934) 180–205.

Waraszkiewicz constructed an uncountable family $W$ of continua in the plane called Waraszkiewicz spirals so that no continuum can be mapped continuously onto every continuum in $W$. Start with the space $X=S^1\cup [1,2]$ and consider the maps $f,g:X\to X$ where

• $f(x)=x$ if $x\in S^1$, $f(x)=\exp(4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $f(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$
• $g(x)=x$ if $x\in S^1$, $g(x)=\exp(-4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $g(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$

Now $W$ is the collection of all inverse limits of all inverse sequences $(X_i,h_i)$ where $X_i=X$ for all $i$ and $h_i\in\{f,g\}$.

For a short and readable description (where I found the construction) see:

W.T. Ingram, Concerning images of continua, Topology Proceedings 16 (1991) 89-93.

This non-existence results has been improved upon a few times, for instance in the following:

S.B. Nadler, The nonexistence of almost continuous surjections between certain continua, Topology and its Applications, 154 (2007) 1008-1014.

There is no such continuum. See

Z. Waraszkiewicz, Sur un problème de M.H. Hahn, Fund. Math. 22 (1934) 180–205.

Apparently, Waraszkiewicz constructs a continuum $X$ so that no continuum can be mapped continuously onto every subcontinuum of $X$. This is extended to a stronger non-existence result in

S.B. Nadler, The nonexistence of almost continuous surjections between certain continua, Topology and its Applications, 154 (2007) 1008-1014.

There is no such continuum. See

Z. Waraszkiewicz, Sur un problème de M.H. Hahn, Fund. Math. 22 (1934) 180–205.

Waraszkiewicz constructed an uncountable family $W$ of continua in the plane called Waraszkiewicz spirals so that no continuum can be mapped continuously onto every continuum in $W$. Start with the space $X=S^1\cup [1,2]$ and consider the maps $f,g:X\to X$ where

• $f(x)=x$ if $x\in S^1$, $f(x)=\exp(4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $f(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$
• $g(x)=x$ if $x\in S^1$, $g(x)=\exp(-4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $g(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$

Now $W$ is the collection of all inverse limits of all inverse sequences $(X_i,h_i)$ where $X_i=X$ for all $i$ and $h_i\in\{f,g\}$.

For a short and readable description (where I found the construction) see:

W.T. Ingram, Concerning images of continua, Topology Proceedings 16 (1991) 89-93.

This non-existence results has been improved upon a few times, for instance in the following:

S.B. Nadler, The nonexistence of almost continuous surjections between certain continua, Topology and its Applications, 154 (2007) 1008-1014.