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# Spanier spaces and covering theory of non-homotopically path Hausdorff spaces

Georgian Mathematical Journal (Impact Factor: 0.34). 05/2011; 20(2). DOI: 10.1515/gmj-2013-0016

Source: arXiv

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Ali Pakdaman, Jun 22, 2015 Available from: Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

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**ABSTRACT:**Let p: X → X/A be a quotient map, where A is a subspace of X. We study the conditions under which p ∗(π 1 qtop (X, x 0)) is dense in π 1 qtop (X/A,∗)), where the fundamental groups have the natural quotient topology inherited from the loop space and p * is a continuous homomorphism induced by the quotient map p. In addition, we present some applications in order to determine the properties of π 1 qtop (X/A,∗). In particular, we establish conditions under which π 1 qtop (X/A,∗) is an indiscrete topological group.Ukrainian Mathematical Journal 05/2014; 65(12):1883-1897. DOI:10.1007/s11253-014-0904-0 · 0.21 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Sormani and Wei proved in 2004 that a compact geodesic space has a categorical universal cover if and only if its covering/critical spectrum is finite. We add to this several equivalent conditions pertaining to the geometry and topology of the revised and uniform fundamental groups. We show that a compact geodesic space X has a universal cover if and only if the following hold: 1) its revised and uniform fundamental groups are finitely presented, or, more generally, countable; 2) its revised fundamental group is discrete as a quotient of the topological fundamental group. In the process, we classify the topological singularities in X, and we show that the above conditions imply closed liftings of all sufficiently small path loops to all covers of X, generalizing the traditional semilocally simply connected property. A geodesic space with this new property is called semilocally r-simply connected, and X has a universal cover if and only if it satisfies this condition. We then introduce a topology on the fundamental group called the covering topology, with respect to which the fundamental group is always a topological group. We establish several connections between properties of the covering topology, the existence of simply connected and universal covers, and geometries on the fundamental group.Topology and its Applications 08/2012; 160(6). DOI:10.1016/j.topol.2013.02.004 · 0.59 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The quasitopological fundamental group $\pi_{1}^{qtop}(X,x_0)$ is the fundamental group endowed with the natural quotient topology inherited from the space of based loops and is typically non-discrete when $X$ does not admit a traditional universal cover. This topologized fundamental group is an invariant of homotopy type which has the ability to distinguish weakly homotopy equivalent and shape equivalent spaces. In this paper, we clarify various relationships among topological properties of the group $\pi_{1}^{qtop}(X,x_0)$ and properties of the underlying space $X$ such as `$\pi_{1}$-shape injectivity' and `homotopically path-Hausdorff.' A space $X$ is $\pi_1$-shape injective if the fundamental group canonically embeds in the first shape group so that the elements of $\pi_1(X,x_0)$ can be represented as sequences in an inverse limit. We show a locally path connected metric space $X$ is $\pi_1$-shape injective if and only if $\pi_{1}^{qtop}(X,x_0)$ is invariantly separated in the sense that the intersection of all open invariant (i.e. normal) subgroups is the trivial subgroup. In the case that $X$ is not $\pi_1$-shape injective, the homotopically path-Hausdorff property is useful for distinguishing homotopy classes of loops and guarantees the existence of certain generalized covering maps. We show that a locally path connected space $X$ is homotopically path-Hausdorff if and only if $\pi_{1}^{qtop}(X,x_0)$ satisfies the $T_1$ separation axiom.04/2013; 10(1). DOI:10.1007/s40062-013-0042-7