Given a set $X$, is there a topology $\tau$ such that the identity $\text{id}_X$ on $X$ is the only continuous injective self-map?
(This is Joel David Hamkins's recent question in the category $\mathbf{Top}^\text{op}$.)
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5It seems to me that Adam Przeździecki's answer at the other question (where we ask for surjectivity) also applies to this question (for sets of size at least continuum), since the Vopěnka-Pultr-Hedrlin result does not require surjectivity. – Joel David Hamkins Aug 18 '15 at 13:42
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Oh - that's right, thanks for your comment. Let's focus on sets with cardinality $< 2^{\aleph_0}$. – Dominic van der Zypen Aug 18 '15 at 14:54