She had difficulty keeping her eyes Open (= because she was very tired) .
她连睁开眼睛的力气都没有了。
He was breathing through his open mouth.
他张着嘴呼吸。
此时形容词open反义词： closed
再看：
He was so sleepy that he could hardly keep his eyes __________.
A. open B. opening C. opened D. to open
此题应选A。容易误选B。主要是受 keep...doing 这一常用句式的影响。
做好本题的关键是要分清：open 表示“开”这一意义时，是形容词还是动词。
1. open用作形容词时，表示状态，意为“开着的”。如：
His eyes were wide open. 他的眼睛睁得大大的。
She had left the door wide open .
她把房门敞开着。
The pass is kept open all the year.
通道一年到头畅通无阻。
The shop isn’t open today. 商店今天不开门。
He was quite open about his reasons for leaving.
他对离开的原因完全未加隐瞒。
He had always been open with her and she always felt she would know if he lied
他对她总是坦诚相待，她也一直觉得如果他说谎，自己一定能察觉。
《柯林斯高阶英汉双解学习词典》
I'm a very open person.
我这个人非常坦诚直率。
2. open用作动词时，表示动作，意为“开”。如：
She opened her arms and gave me a big hug.
她张开双臂，紧紧地拥抱了我。
《柯林斯高阶英汉双解学习词典》
Open your mouth and say "ah". 张开嘴说“啊”。
Please open the window. 请打开窗户。
What if the parachute doesn't open?
降落伞打不开怎么办？
The flowers are starting to open.
花开始绽放了。
open用作动词，是终止性动词，通常不能用来表示“开”这一动作的持续。如。如：
这门打开已有一周了。
正：The door has been open for a week.
误：The door has been opened for a week.
晚上让窗户开着。
正：Keep the window open at night.
误：Keep the window opening at night.
keep sb(sth) doing sth.使某人或某物不定地做某事（动作之间稍有间隔）
通过以上分析，我们可以清楚地知道：上面一题应用keep...open, 而不能用keep...opening, 其道理已很显然。假若我们要用keep...opening, 那其实就是要使open 这一终止性动作不断地反复发生，这显然是荒唐可笑的。
顺便说一下：
keep one’s ears/eyes open (for sth)
（对…）保持警觉；注意；留心
to be quick to notice or hear things
