Question

Let $LaTeX: \left(X,\mathscr{T}\right)$ be a topological space, let and be paths in such that . Show that $LaTeX: f_1\ast f_2: I\to X$ defined by $LaTeX: f_1\ast f_2(x)=\begin{cases} f(2x)&\text{ if }0\le x\le \frac{1}{2}\\ f(2x-1)&\text{ if }\frac{1}{2}\le x\le 1 \end{cases}$ is a path in

Answer 1

12(2x+3;45451 (8,0T) is a topological Space. f, fa: I are path inx, sit f (0 = faroo. h*fa: Ix as 4* fz (us= {f(2x): 0575/ clearly for osxc Y .f is continuous 8 fz is continuo on Ž<xsl. So, fix fz is continuous on I k%} To show: f*fz is coolfinuous at x = 42 For every nbhd u of fix fo /₂) = fill=f2(0), there are nibhos. V e v² of Y₂ set fi(v) cu 8 farva cu Take i= 1² nv².. Then / FV ( 2 / Ev's & Evy . a (vot, 8t are (5+ f2) (V) = fi lv') o felvy CU: So, I V st £cus (fitto) (V) CO. - Hence ha fz is continuous at sc =% Therefore fpt fz is continuous on I. Hence first fa is path inx - continuous atx=42) .