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Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa: Are finite spaces a model for finite CW-complexes?

So, when it comes to computing homotopy groups of finitely triangulated spaces there is really no loss of generality.

One might try draw as an exercise a finite topological space having the same homotopy groups as the 2-sphere: it will be a set of cardinality $14$, namely the poset (order given by inclusion) of all simplices of the boundary of the $3$-simplex with the order topology.

Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa: Are finite spaces a model for finite CW-complexes?

So, when it comes to computing homotopy groups of finitely triangulated spaces, there is really no loss of generality.

One might try to draw as an exercise a finite topological space having the same homotopy groups as the 2-sphere: it will be a set of cardinality $14$, namely the poset (order given by inclusion) of all simplices of the boundary of the $3$-simplex with the order topology.

Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa: Are finite spaces a model for finite CW-complexes?

So, when it comes to computing homotopy groups of finitely triangulated spaces there is really no loss of generality.

One might draw as an exercise a finite topological space having the same homotopy groups as the 2-sphere: it will be a set of cardinality $14$, namely the poset (order given by inclusion) of all simplices of the boundary of the $3$-simplex with the order topology.

Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa: Are finite spaces a model for finite CW-complexes?

So, when it comes to computing homotopy groups of finitely triangulated spaces there is really no loss of generality.

One might try draw as an exercise a finite topological space having the same homotopy groups as the 2-sphere: it will be a set of cardinality $14$, namely the poset (order given by inclusion) of all simplices of the boundary of the $3$-simplex with the order topology.

Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa: Are finite spaces a model for finite CW-complexes?

Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa: Are finite spaces a model for finite CW-complexes?

So, when it comes to computing homotopy groups of finitely triangulated spaces there is really no loss of generality.

One might draw as an exercise a finite topological space having the same homotopy groups as the 2-sphere: it will be a set of cardinality $14$, namely the poset (order given by inclusion) of all simplices of the boundary of the $3$-simplex with the order topology.