For $X$ and $Y$ path connected, if $X$ and $Y$ are homotopy equivalent, is there base point $x \in X$ and $y \in Y$ such that $(X,x) \simeq (Y,y)$?
117877
Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces?
66548
proving topological equivalence
85242
Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$
19984
Example of topological spaces with continuous bijections that are not homotopy equivalent
142698
Proving that the fundamental groups of two spaces with same homotopy type are isomorphic
127826
Is the geometric realization of a pointed category contractible?
94059
Extending a Homotopy from $X \times [0,1]$ to $X \times \mathbb{R}$
108226
Are these bases for a topology?
126192
Connected, locally connected, path-connected but not locally path-connected subspace of the plane
71037
Is exponential map locally a diffeomorphism w.r.t. base point?
18828
Base and empty set of a topology
131155
$x\sim y\iff x^2\equiv y^2\pmod{\!6}$ is an equivalence relation.
108909
Is the unit sphere in $\Bbb R^4$ path connected?
156443
if $x$ and $y$ commute with $[x y]$, then $[x y]$ is nilpotent
157920
Can I define $y=y(x)$?
84654
The graph of xy = 1 is connected or not
106443
Question on relative homology