With out the axiom of choice is the following true: If a topology on $X$ contains every infinite subset of $X$, then this is the discrete topology.
109374
Is the Discrete Topology on $X$ the Only One Containing All Infinite Subsets of $X$?
481
Can compact sets completey determine a topology?
65308
Axiom of Choice: Family of non-empty sets mutually disjoint or not?
33563
Is the axiom of choice used in the proof that every open set is the union of basis elements (Mukres, Lemma 13.1)
135025
Axiomatization of a Topology using the Boundary
22240
Every infinite subset of a countable set is countable.
58447
Is axiom of choice required for there to be an infinite linearly independent set in a (non-finite-dimensional) vector space?
31617
Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?
73336
If $A$ is a nowhere dense subset of a topological space $X$, then $X\setminus A$ is dense in $X$.
130865
Why is the axiom of choice separated from the other axioms?
118765
A topological space is locally compact then here is an open base at each point has all of its set with compact closure
44974
Infinite set and countable subsets
98580
Lebesgue-measurable sets requiring the Axiom of Choice to construct
100476
Why does a Hausdorff but not countably compact space have an infinite closed discrete subset?
77563
Homeomorphism between spaces equipped with cofinite topologies
21202
Axiom of choice is equivalent to every relation includes a function with the same domain
152272
Can you explain the "Axiom of choice" in simple terms?