Let $X$ be a topological space, and let $x$ be a point of $X$. By a neighborhood of $x$ I mean any subset $N$ of $X$ such that $x \in A \subseteq N$ for some open set $A$ of $X$. A topological space is said to be simply connected if it is path-connected and has a trivial fundamental group. Now, consider the following properties:
(1) for any neighborhood $N$ of $x$ there exists an open simply connected neighborhood $U$ of $x$ such that $U \subseteq N$;
(2) for any neighborhood $N$ of $x$ there exists a simply connected neighborhood $U$ of $x$ such that $U \subseteq N$;
(3) there exists some open simply connected neighborhood $U$ of $x$;
(4) there exists some simply connected neighborhood $U$ of $x$;
(5) for any neighborhood $N$ of $x$ there exists some open neighborhood $U$ of $x$ such that $U \subseteq N$ and any loop in $U$ based at $x$ is nullhomotopic in $X$;
(6) for any neighborhood $N$ of $x$ there exists some neighborhood $U$ of $x$ such that $U \subseteq N$ and any loop in $U$ based at $x$ is nullhomotopic in $X$;
(7) there exists some open neighborhood $U$ of $x$ such that any loop in $U$ based at $x$ is nullhomotopic in $X$;
(8) there exists some neighborhood $U$ of $x$ such that any loop in $U$ based at $x$ is nullhomotopic in $X$;
(9) for every neighborhood $N$ of $x$ there exists an open path-connected neighborhood $U$ of $x$ such that $U \subseteq N$ and every loop in $U$ is nullhomotopic in $N$;
(10) for every neighborhood $N$ of $x$ there exists a path-connected neighborhood $U$ of $x$ such that $U \subseteq N$ and every loop in $U$ is nullhomotopic in $N$.
For $n=1,\dots,10$, let $P_n$ denote the statement "property (n) holds for every $x \in X$". Now $P_1,\dots,P_{10}$ are all meaningful localizations of the notion of simple connectedness. Actually, $P_1$ is the definition of locally simply connected space accepted by most of the authors, while properties $P_7$ or $P_8$ are usually taken as the definition of semilocally simply connected space (this notion seems to be by far the most relevant because of its applications in the theory of covering spaces). Some authors define a space to be locally simply connected if it satisfies $P_3$ (see e.g. Knapp, Lie Groups and Beyond) or $P_4$ (see e.g. Chevalley, Theory of Lie Groups I), while $P_{10}$ is part of a hierarchy of definitions of locally n-connected spaces, usually denoted as $LC^n$ (see e.g. Sakai, Geometric Aspects of General Topology, where the definition of $LC^1$ space looks different from $P_{10}$, but it is completely equivalent to it: see Section 4.14 of that book and especially Proposition 4.14.6). From some research I made in the literature, it seems to me that topologists did not pay much attention to a careful study of the notion of locally simply connected space, contrary to what they did for locally connected or locally path-connected spaces.
My aim is to give a full picture of the relative strength of these various definitions. In the notes below I collect all the results I got. However, I could neither prove nor disprove the following implications:
$\bullet P_2 \Rightarrow P_i$ for $i=1$ or $3$;
$\bullet P_{10} \Rightarrow P_i$ for $i=1,2,3$ or $4$.
Does anybody have any idea about possible proofs of these implications or any counterexample to them?
Thank you very much in advance for your kind attention.
NOTE 1. The following implications immediately follow from the definitions:
$\bullet$ $(1) \Rightarrow (2) \Rightarrow (4) \Rightarrow (6)$;
$\bullet$ $(1) \Rightarrow (3) \Rightarrow (4)$;
$\bullet$ $(5) \Leftrightarrow (6) \Leftrightarrow (7) \Leftrightarrow (8)$;
$\bullet$ $(1) \Rightarrow (9) \Rightarrow (7)$;
$\bullet$ $(2) \Rightarrow (10) \Rightarrow (8)$;
$\bullet$ $(9) \Rightarrow (10)$.
So we have:
$\bullet$ $P_1 \Rightarrow P_2 \Rightarrow P_4 \Rightarrow P_6$;
$\bullet$ $P_1 \Rightarrow P_3 \Rightarrow P_4$;
$\bullet$ $P_5 \Leftrightarrow P_6 \Leftrightarrow P_7 \Leftrightarrow P_8$;
$\bullet$ $P_1 \Rightarrow P_9 \Rightarrow P_7$;
$\bullet$ $P_2 \Rightarrow P_{10} \Rightarrow P_8$;
$\bullet$ $P_9 \Rightarrow P_{10}$.
NOTE 2. In general $P_3 \nRightarrow P_{10}$ (and so $P_3 \nRightarrow P_{2}$), even if we make the additional assumption that $X$ is path connected and locally path-connected. To see this, consider the following example (given in Fulton, Algebraic Topology). For any positive integer $n$, let $C_n$ be the circle in the plane $\{ (x,y,z) \in \mathbb{R}^3: z=0\}$ with center $(1/n,0,0)$ and radius $1/n$. Put $C= \bigcup_{n=1}^{\infty} C_n$ and let $X$ be the cone with vertex $(0,0,1)$ and base $C$, that is $X$ is the union of all line segments from $(0,0,1)$ to points in $C$. Usually $X$ is called a clamshell. Then $X$ is simply connected, so it satisfies $P_3$, but it does not satisfy condition (10) in $(0,0,0)$.
NOTE 3. In general $P_4 \nRightarrow P_3$, even if we make the additional assumption that $X$ is path connected and locally path-connected. To build such a space, let $S_1$ be the clamshell defined in Note 2. Define $S_{2}$ as the image of $S_1$ under the translation of vector $(0,0,-1)$, and let $S= S_1 \cup S_2$. Now imagine to deform the segment of end-points $(0,0,-1)$ and $(0,0,1)$ into a circle $C$ by joining its end-points, so that the clamshells now wrap around $C$. Let $X$ be the space so obtained (more formally, $X$ is the quotient space obtained from $S$ by identifying the points $(0,0,-1)$ and $(0,0,1)$). $X$ is path-connected, locally path-connected and satisfies $P_4$, but it does not satisfy $P_3$. Indeed, if $x$ is (the image under our deformation of) the vertex of a clamshell and $N$ is a simply connected open neighborhood of $x$, then you can easily convince yourself that $N$ should contain all of $C$, which in turn would imply that $N$ is not simply connected, a contradiction.
NOTE 4. In general $P_5 \nRightarrow P_4$, even if we make the additional assumption that $X$ is path connected and locally path-connected. To build such a space we use the same trick as before. Let $S$ be the clamshell defined in Note 2, and imagine to deform the segment of end-points $(0,0,0)$ and $(0,0,1)$ into a circle $C$ by joining its end-points, so that the clamshell now wraps around $C$. Let $X$ be the space so obtained (again, formally $X$ is the quotient space obtained from $S$ by identifying the points $(0,0,-1)$ and $(0,0,1)$). $X$ is path-connected, locally path-connected and satisfies $P_5$, but it does not satisfy $P_4$. Indeed, if $x$ is (the image under our deformation of) the vertex of the clamshell and $N$ is a simply connected neighborhood of $x$, then you can easily convince yourself that $N$ should contain all of $C$, which in turn would imply that $N$ is not simply connected, a contradiction.
NOTE 5. In general $(2) \nRightarrow (3)$. To see this is the case, let e.g. $B$ be the infinite union of decreasing closed infinite brooms in $\mathbb{R}^2$ as illustrated in the picture of the post Show that is not locally connected at p, with $p=(0,0)$ and $a_1=(1,0)$, and let $C= \bigcup_{n=1}^{\infty} C_n$, where $C_n$ is the circle of center $(1+1/n,0)$ and radius $1/n$. Then take $X=B \cup C$ and $x=p$.
NOTE 6. The space $B$ defined in the previous note shows that in general $(10) \nRightarrow (9)$. Anyhow, we have $P_{10} \Rightarrow P_9$. Indeed, $P_{10}$ implies that $X$ is locally path-connected (see e.g. Munkres, Topology, § 25, Exercise 6), so that if $x \in X$, $N$ is a neighborhood of $x$, and $U$ is a path-connected neighborhood of $x$ contained in $N$ such that every loop in $U$ is nullhomotopic in $N$, then there exists an open path-connected neighborhood $V$ of $x$ contained in $U$, and clearly every loop in $V$ is nullhomotopic in $N$. So we have $P_9 \Leftrightarrow P_{10}$.
NOTE 7. This question has now been posted on mathoverflow.net as Definition of Locally Simply Connected Space.
