What can be said about topological properties of a space $X$ that is homeomorphic to $X\times X$ $?$
46771
$X$ is homeomorphic to $X\times X$ (TIFR GS $2014$)
17165
Is total boundedness a topological property?
51871
Characterization of Topology
131082
Proof that $\mathbb R/[0,1]$ and $\mathbb R$ are homeomorphic
46970
If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism.
149605
Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?
14713
$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology Then
158221
Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?
148213
Given a topological space $X$, why are both $X$ and $\emptyset$ open and closed?
135025
Axiomatization of a Topology using the Boundary
77563
Homeomorphism between spaces equipped with cofinite topologies
57816
$(S^1 \times [0,1])/\sim$ homeomorphic to unit disk $ D^2$
42913
Equivalence of the definition of the Subbasis of a Topology
115173
The Cantor set is homeomorphic to infinite product of $\{0,1\}$ with itself - cylinder basis - and it topology
79213
Proof about a Topological space being arc connect
65034
Are continuous self-bijections of connected spaces homeomorphisms?
31617
Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?