Cut off any frayed edges of the chair fabric damaged by the cat.
Apply fabric glue, or fray check, over the edges of the tear.
Choose thread that matches the chair fabric and thread it through a needle. Make sure to tie a small knot at the end of the thread.
Fold the edges of the tear inward and sew the folds so they will not come undone.
Sew the folded edges together. Using a running stitch will be the most transparent. Fasten the thread about 1/4-inch from the start of the tear and make small stitches by passing the needle in and out of the fabric in a straight line.
Cut away any remaining thread after fully closing the tear.