### Abstract

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) Π_{1}((X, x) ⩗ (X, x), x) is naturally isomorphic to the free product Π_{1}(X, x)*Π_{1} (X, x); (3) Π_{1}((CX, x)V(CX, x), x) is trivial. There exists a simply connected, locally simply connected Tychonoff space X with x ∈ X, such that (X, x) ⩗ (X, x) is not simply connected.

Original language | English |
---|---|

Pages (from-to) | 239-249 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 116 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1992 |

Externally published | Yes |

### Fingerprint

### Keywords

- Cone
- First countable
- Fundamental group
- Locally simple
- One point union

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**A locally simply connected space and fundamental groups of one point unions of cones.** / Eda, Katsuya.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 116, no. 1, pp. 239-249. https://doi.org/10.1090/S0002-9939-1992-1132409-0

}

TY - JOUR

T1 - A locally simply connected space and fundamental groups of one point unions of cones

AU - Eda, Katsuya

PY - 1992

Y1 - 1992

N2 - 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) Π1((X, x) ⩗ (X, x), x) is naturally isomorphic to the free product Π1(X, x)*Π1 (X, x); (3) Π1((CX, x)V(CX, x), x) is trivial. There exists a simply connected, locally simply connected Tychonoff space X with x ∈ X, such that (X, x) ⩗ (X, x) is not simply connected.

AB - 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) Π1((X, x) ⩗ (X, x), x) is naturally isomorphic to the free product Π1(X, x)*Π1 (X, x); (3) Π1((CX, x)V(CX, x), x) is trivial. There exists a simply connected, locally simply connected Tychonoff space X with x ∈ X, such that (X, x) ⩗ (X, x) is not simply connected.

KW - Cone

KW - First countable

KW - Fundamental group

KW - Locally simple

KW - One point union

UR - http://www.scopus.com/inward/record.url?scp=84966211913&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966211913&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1992-1132409-0

DO - 10.1090/S0002-9939-1992-1132409-0

M3 - Article

VL - 116

SP - 239

EP - 249

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -