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The question was motivated by this question of Anton Petrunin. By a metric continuum we understand a connected compact metric space. Let $p$ be a positive real number. A metric continuum $X$ is call …

The answer to this question is positive. A required path $\gamma$ can be constructed inductively using the following Lemma. For any continuum $P\subset\mathbb R^2$, distinct points $x,y\in P$, and $\ …

No solenoid can be embedded into the Mobius strip. To derive a contradiction, assume that some solenoid $S$ embeds into the Mobius strip $M$. Let $\pi:C\to M$ be a 2-fold covering map of the cylinder …

Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim_\varepsilon y$ if there exists a sequence $C_1,\dots,C_n$ of connected subsets of …

This problem was motivated by the MO problems: "Running most of the time in a connected set", "Is every metric continuum almost path connected?" and "Are $\varepsilon$-connected components dense?". L …

Definition. A closed subset $S$ of a topological space $X$ is called a separator between points $x,y\in X\setminus S$ if the points $x$ and $y$ belong to different connected components of $X\setminus …

A graph is called $n$-connected if it remains connected after removal $\le n$ vertices. Question. What is the name of an analogous property of topological spaces: a space that remains connected after …