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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 …

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 …

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 …

$p^{\ast}$ may not be surjective without extra assumptions. For example, you can check out Zeeman's example in Spanier's textbook of non-equivalent coverings corresponding to the same subgroup of $\pi …

$F=S^2\backslash U$ 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 $U$ …

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 …

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 …

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))\ …

Perhaps this is helpful for the locally compact case. Corollary 3.1.12 in "Topological groups and related structures" by Arhangel'skii and Tkachenko gives a nice characterization of connectedness in …

The space you are talking about is sometimes called the Griffiths space. As Henry Horton suggests, to prove it is not contractible you can show the fundamental group is non-trivial (even though its Ce …

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 …

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 …

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 …

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 …