English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let CX be the cone over a space X. Let a space X be first countable at x, then the following are equivalent: (1) X is locally simply connected at x; (2) ir1 ((X, x) V (X, x), x) is naturally isomorphic to the free product r I(X, x) * r1(X, x); (3) r I((CX, x) V (CX, x), x) is trivial. There exists a simply connected, locally simply connected Tychonoff space X with x e X, such that (X, x) V (X, x) is not simply connected. Griffiths [2] proved Theorem 1 (H. B. Griffiths). Let a space X be locally simply connected at x E X and also first countable at x. Then, for an arbitrary space Y with y E Y, the fundamental group 7r I((X, x) v (Y, y), x) of the one point union (X, x) V (Y, y) is naturally isomorphic to the free product 7r1 (X, x) * 7r,(Y, y). In addition, in the same paper he proved that local simple connectivity is essential in this theorem. On the other hand the author [1] of the present paper has shown that the first countability in the theorem is also essential. In ? 1 we shall prove Theorem 2. Let spaces X, Y be first countable at x, y, respectively. Then the following are equivalent: (1) At least one of X and Y is locally simply connected at x or y, respectively; (2) 7rI((X, x) V (Y, y), x) is naturally isomorphic to the free product rl (X, X) * 7fl(Y, Y) ; (3) 7rI ((CX, x) V (CY, y), x) is trivial, where CX, CY are the cones over X, Y, respectively. Corollary 3. Let a space X be first countable at x. Then the following are equivalent: (1) X is locally simply connected at x; (2) 7r1 ((X, x) V (X, x), x) is naturally isomorphic to the free product ilr(X, x) * 7rl(X, x); (3) r I((CX, x) V (CX, x), x) is trivial. Received by the editors February 27, 1990 and, in revised form, November 26, 1990. 1991 Mathematics Subject Classification. Primary 55Q20, 55Q52.

A locally simply connected space and fundamental groups of one point unions of cones