On the homotopy groups of separable metric spaces

The n-dimensional Spanier group

“…The space X is called n-homotopically Hausdorff if X is n-homotopically Hausdorff at x, for every x ∈ X. See [9] for more details. (5) Let x 0 , x 1 be two points of X, σ : [0, 1] → X be a path from σ(0) = x 0 to σ(1) = x 1 , and [α] ∈ π n (X, x 1 ).…”

Compactly generated quasitopological homotopy groups with discontinuous multiplication

“…It is an open problem to understand when π Q (X, p) is or is not a topological group with the standard operations. For example, is π Q (X, p) always a topological group if Q ≥ 2 (Problem 5.1 [1], abstract [6])? If Q ≥ 1 must π Q (X, p) be a topological group if X is a pathconnected continuum and π Q (X, p) is compactly generated?…”

“…In general the topology of π Q (X, p) is an invariant of the homotopy type of the underlying space X, π Q (X, p) is a quasitopological group (i.e. multiplication is continuous separately in each coordinate and group inversion is continuous), and each map f : X → Y induces a continuous homomorphism f * : π Q (X, p) → π Q (Y, f (p)) [5].…”

“…If X has strong local properties (for example if X is locally nconnected for all 0 ≤ n ≤ Q) then π Q (X, p) is discrete [3], [2], [5], and hence a topological group.…”