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Andrew's answer is right, but I'll just throw in a few comments since "topological" homotopy invariants are of great interest to me. Here Paul Fabel has shown that $\pi_{1}^{top}$ on the Hawaiian earr …

There is no such continuum. See Z. Waraszkiewicz, Sur un problème de M.H. Hahn, Fund. Math. 22 (1934) 180–205. Waraszkiewicz constructed an uncountable family $W$ of continua in the plane called Wa …

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 …

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 …

This answer takes a more general viewpoint than Alexandre's. The generality is in response to the small number of assumptions on the spaces involved. First, you should assume that $Y$ is locally path …

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 …

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 …

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 …

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 …

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 …

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 …

A space $X$ whose topology agrees with the final topology with respect to all maps $I\to X$ is often called a delta-generated ($\Delta$-generated) space. The category of $\Delta$-generated spaces is a …

Suppose you have basepoints $x_0\in X$, $z_0\in Z$ and $p(z_0)=f(x_0)$. The lift $\tilde{f}:X\to Z$ such that $p\circ \tilde{f}=f$ exists and is continuous if and only if 1) $f_{\ast}(\pi_1(X,x_0))\ …

One of the easiest examples is the rational numbers with the subspace topology of the real line with the K-topology. Total path disconnectedness is not entirely necessary for multiplication of $\pi_{1 …

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 …