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Let $G_n$ be your "grid" in $\mathbb{R}^n$ for $n\geq 2$. Let $p_{n,n-1}:G_n\to G_{n-1}$ be the usual projection map. Then $X$ can be identified canonically with the inverse limit $\varprojlim_{n}(G_n …

The category of $\Delta$-generated spaces is quite large as it includes all first countable, locally path connected spaces. Hence, all compact, connected, locally connected subsets of $\mathbb{R}^n$ a …

Let $X$ be a a "topological tree" by your definition. Then $X$ is uniquely arcwise connected and Hausdorff. Let $f:S^n\to X$ be a map from the $n$-sphere where $n\geq 1$. It follows from the Hahn-Mazu …

No, the harmonic archipelago (An illustration is on pg 7 of W. A. Bogley and A. J. Sieradski, Universal path spaces, preprint) is a locally path connected subspace of $\mathbb{R}^{3}$ and has uncounta …

In the Griffiths Twin Cone (or double cone over the shrinking wedge of circles), $G\subseteq \mathbb{R}^3$, all injective loops are null-homotopic yet $\pi_1(G)$ is uncountable. Hence, injective loops …

Here is the key step you need to finish the proof: We are supposing $X$ is locally path-connected and semilocally simply connected, $\pi:P(X,x_0)\to \widetilde{X}$ is the quotient map identifying path …

The figure below gives a simple but extreme counterexample, which I think has all the lifting properties one might want except for actually being a true fibration. The map is the identity everywhere e …

My answer to this question gives an example of a locally path connected (but non-semilocally simply connected) space $HA\subset\mathbb{R}^3$ called the Harmonic archipelago: draw the Hawaiian earring …

$F=S^2\backslash U$ need not be path connected, e.g. $F$ could be homeomorphic to the closed topologists sine curve. However, every path component of $F$ must be simply connected. By identifying $U$ …

Thanks to the comments, my original cardinality bound $\aleph_1\leq |S|\leq \mathfrak{c}$ has been refined to the equality $|S|=\mathfrak{c}$ that I originally suspected. For Question 1: $S$ has the …

A space $X$ is called $\Delta$-generated if $U$ is open in $X$ if and only if $\alpha^{-1}(U)$ is open in $[0,1]$ for every path $\alpha:[0,1]\to X$. It's easy to see that a space $X$ is $\Delta$-gen …

Fundamental Group: The fundamental group of a planar set naturally injects into the first Cech homotopy group, which is an inverse limit of free groups. In particular, the algebraic restrictions gaine …

One-dimensional metric spaces and planar sets do have the property that you're interested in. To explain why this works out in such generality requires a combination of planar topology, continuum theo …

I contacted Juraj Činčura and he kindly wrote back and directed me to the following observation that is a consequence of results in the paper that David White noted in his answer. Cartesian closed co …

No such homomorphism is possible. The prototypical nature of the Griffiths twin cone guarantees this. Let $X,Y$ be contractible with basepoints $x,y$ respectively. We only need to assume $\{x\}$ and …