On the Topological Fundamental Groups of Quotient Spaces

Abstract

Abstract. Let p : X → X/A be a quotient map, where A is a subspace of X. We explore conditions under which p * (π qtop 1

On locally 1-connectedness of quotient spaces and its applications to fundamental groups

“…J. Brazas [4] introduced a new topology on π 1 (X, x) coarser than the topology of π qtop 1 (X, x) and proved that fundamental groups by this new topology are topological groups, denoted by π τ 1 (X, x). Since the topology of π τ 1 (X, x) is coarser than π qtop 1 (X, x), the results of [10] remain true and we have a similar result to Theorem B for π τ 1 (X, x) and π τ 1 (X/(A 1 , ..., A n ), * ).…”

“…For a topological space X, a loop α in X based at x is called semi-simple if α −1 ({x} c ) = (0, 1) and is called 1]. Also, every geometrically simple loop is homotopic to a semi-simple loop [10].…”

“…Theorem 2.4. [10] Let A 1 , A 2 , ..., A n be disjoint path connected, closed subsets of a first countable, connected, locally path connected space X such that X/(A 1 , A 2 , ..., A n ) is semi-locally simply connected, then for each a ∈ n i=1 A i , p * : π 1 (X, a) −→ π 1 (X/(A 1 , A 2 , ..., A n , * ) is an epimorphism.…”

“…It is known that this construction gives rise a homotopy invariant functor π qtop 1 applying the above functor on the quotient map p : X −→ X/ ∼, we have a continuous homomorphism p * : π qtop 1 (X, x) −→ π qtop 1 (X/ ∼, * ). Recently, the authors [10] proved that if X is a first countable, connected, locally path connected space and A 1 , A 2 , ..., A n are disjoint path connected, closed subspaces of X, then the image of p * is dense in π qtop 1 (X/(A 1 , A 2 , ..., A n )). By using the results of [10] and Theorem A we have the primary application of the main result of this paper as follows:…”

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

“…The authors [12] proved that for a first countable, simply connected and locally path connected space X, if A ⊆ X is a closed path connected subset, then the quotient space X/A has indiscrete topological fundamental group. Therefore by Corollary 5.2 we have the following theorem that gives a family of Spanier spaces.…”