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No. Let $X$ be the disjoint union of the real line and one isolated point. The quotient by "movability" collapses the real line to one point, so the quotient is a discrete space of two points, which …

In my answer to https://mathoverflow.net/questions/26414 I descibed a somewhat simpler-looking example than Nik's, but proving that it works may be harder. Take two copies of $\beta\mathbb N$ and glu …

Let $\{X,Y\}$ be a partition of the union of the family $S$, and let p be a point in the intersection of $S$ (which you've assumed is nonempty). Without loss of generality, p is in $X$. But, since a …

All three notions, "countably tight," "sequential," and "Frechet-Urysohn," say that each point $p$ in the closure of a set $A$ can be "approached in some countable way" by points from $A$. The differ …

Non-principal ultrafilters with the property in the question for all partitions are called "selective" or "Ramsey" ultrafilters. The "Ramsey" terminology comes from the following connection, due to K …

For Q3, the answer is yes. Think of the Cantor set as consisting of the points that have a base-3 expansion containing only 0's and 2's. Now take the subset of those points where the 2's occur only …

No. Consider the space whose points are all sequences of 0's and 1's of length $\leq\omega$. Visualize it as the binary tree plus "limits" for all paths through the tree, and topologize it accordingl …

It is not possible to have two topologies on $X$ with the same nets converging to the same points. To prove it, consider any two distinct topologies $T$ and $T'$ on $X$, and suppose, without loss of g …

In the category of topological spaces, you can't in general even cancel the two-point discrete space. The only counterexample I know is a bit complicated though: Start with the disjoint union of two …

Let the complete bounded metric space be the unit ball of the Hilbert space $l_2$, and let $\{e_i:i\in\mathbb N\}$ be an orthonormal basis. Consider the following sequence $(x_n)$, which goes from ea …

As Joel David Hamkins commented, this is likely to be a cardinal characteristic. Let me define (I hope temporarily) the relevant characteristic $\mathfrak{dfpmk}$ (named after the OP and the author o …

Since $(\mathbb N-\{0\})^2$ is a 2-sided ideal in $\mathbb N^2$, it follows that $\beta((\mathbb N-\{0\})^2)$ is a 2-sided ideal in $\beta(\mathbb N^2)$ and therefore contains all of the latter semigr …

Although the question has already been thoroughly answered, it might be worthwhile to point out that the result can be separated into the combinatorial "meat", which doesn't involve ultrafilters, and …

Let $X$ be an infinite set. Take the uniformity $\mathcal U$ generated by those equivalence relations on $X$ that have only finitely many equivalence classes. The induced topology on $X$ is discrete …

Joel has completely answered the question, but let me add another example, with a bigger failure of injectivity of $g$. Let $D$ be the set of points in the plane of the form $(\frac1n,\sin n)$ for po …