FRAGMENT
Embroidery on Knotted Net piece
Italian, from the 16th century
http://www.metmuseum.org/works_of_art/collection_database/european_sculpture_and_decorative_arts/fragment/objectview_altviewzoom.aspx?page=7&sort=0&sortdir=asc&keyword=&fp=1&dd1=12&dd2=0&vw=1&collID=12&oID=120029990&vT=1&iID=2&iPage(14)&hi=0&ov=0

PILLOW CASE
Embroidery on knotted net
German, from the 18th century 
http://www.metmuseum.org/works_of_art/collection_database/european_sculpture_and_decorative_arts/pillow_case/objectview_enlarge.aspx?page=1570&sort=0&sortdir=asc&keyword=&fp=1&dd1=12&dd2=0&vw=1&collID=12&OID=120045460&vT=1&hi=0&ov=0

FRAGMENT
This looks like Embroidery on Tulle (Tulle was a very fine, square mesh, knotted net) with appliqués.
Is it knotted (Shuttle) or is it twisted (bobbin)?  The picture can not show clearly if a knot is present. 
http://www.metmuseum.org/Works_of_Art/collection_database/european_sculpture_and_decorative_arts/lace_fragment//objectview_altviewzoom.aspx?page=1&sort=0&sortdir=asc&keyword=&fp=1&dd1=12&dd2=0&vw=1&collID=12&OID=120036154&vT=1&iID=1&iPage=1&hi=0&ov=0

FRAGMENT
Italian, from the 16th century
http://www.metmuseum.org/Works_of_Art/collection_database/european_sculpture_and_decorative_arts/fragment//objectview.aspx?OID=120029990&collID=12&dd1=12

PILLOW SHAM
American, ca 1862
Filet Guipure technique

http://www.metmuseum.org/works_of_art/collection_database/american_decorative_arts/pillow_sham/objectview_zoom.aspx?page=368&sort=0&sortdir=asc&keyword=&fp=1&dd1=1&dd2=0&vw=1&collID=1&OID=10014371&vT=1&hi=0&ov=0

CAP CROWN
French, from the 18th century
It looks to be embroidery on a square mesh.  Would it be knotted Tulle? or Fond de Tulle (Bobbin Lace)?
http://www.metmuseum.org/works_of_art/collection_database/european_sculpture_and_decorative_arts/cap_crown/objectview_enlarge.aspx?page=620&sort=0&sortdir=asc&keyword=&fp=1&dd1=12&dd2=0&vw=1&collID=12&OID=120036520&vT=1&hi=0&ov=0