relative homotopy under composition

1. Proposition

Let 20230802-relative_homotopy_under_composition_3e3122d320f7bffebdc8b0f35f367104b8445b3e.svg, 20230802-relative_homotopy_under_composition_c369ec44162417e211640a8ba58bff3935984253.svg and 20230802-relative_homotopy_under_composition_af38f926f7f15df7115d86b1debf0d15907543ab.svg be topological spaces 20230802-relative_homotopy_under_composition_13c81751d7f2da444e8853c50a8e3b7b539eecc9.svg, 20230802-relative_homotopy_under_composition_df7988e88414e060187b18250c529a1e33f1a33b.svg, 20230802-relative_homotopy_under_composition_a7ac6e29420c9e78508bd197f9946dbb49cd4453.svg and 20230802-relative_homotopy_under_composition_261c3bf9fb709eadf49e1d8fb7f5676e6928601d.svg continuous map such that

20230802-relative_homotopy_under_composition_31dbaa56c0c32782dfa5942a9d553dfe52b978ad.svg

and 20230802-relative_homotopy_under_composition_d2c1a6c87a6528da34e4927338f9cd122c5057a3.svg Then

20230802-relative_homotopy_under_composition_13f9966a7b9ed28440dbe07656d950e6b404a0da.svg

2. Proof

By assumption, there exist homotopies 20230802-relative_homotopy_under_composition_1d2170c8abeede710ec1bb19ab8a970468a81da5.svg and 20230802-relative_homotopy_under_composition_830a372394f824460544f6afea10cd27dfdad7e1.svg such that 20230802-relative_homotopy_under_composition_830fc4a685f5d5214bec2d5cdbdf0d9b36ce29c4.svg for 20230802-relative_homotopy_under_composition_959bd8579ea046b8fbd04795a75ba0b62f7d8161.svg Then we define

20230802-relative_homotopy_under_composition_f10bdcd5b98da2b938b8a9bc06be2ee769010893.svg

the map is as a composition of continuous maps itself continuous (see: Stetigkeit der Diagonalabbildung)

20230802-relative_homotopy_under_composition_8678ffaaf3dbfeefb2a2f6806af03adb67124a8c.svg

Furthermore

20230802-relative_homotopy_under_composition_ed4c80c9a5cc9aee00fad39d06c30f4032c9da2b.svg

Let 20230802-relative_homotopy_under_composition_83af3dec20aa987a651b514cef58029da7f94cda.svg and 20230802-relative_homotopy_under_composition_959bd8579ea046b8fbd04795a75ba0b62f7d8161.svg then since 20230802-relative_homotopy_under_composition_707e4db28af2d310fe0bae091653af1911ebc41c.svg, we conclude

20230802-relative_homotopy_under_composition_124479873eea7fa7fa2cc785f0226ca198f15d72.svg

Date: nil

Author: Anton Zakrewski

Created: 2024-10-13 So 15:38