A deaf, 21-year-old mother sits in an office in the Pitts Hotel with her 4-month-old baby nestled in her lap. She has been turned out of her parents' home after an argument with her father about who will cash her government assistance check. She and her son will spend Christmas on the "open market" with the others, mostly women and children, for whom the city has no space in its shelters for the homeless.

Despite the fact that the city is obtaining a record amount of shelter space for families -- including contracts with two new private shelters that opened this fall -- increased evictions, the shortage of low-income housing and an influx of poor families means the city must turn to motels, or the "open market," every night.

These open-market mothers are given bus tokens and are sent chiefly to the Capital City and Hayes motels in Northeast, which are 35 minutes and two transfers distant, for one night only.

The next day, they return by bus to the Pitts Hotel at 14th and Belmont streets NW for city-provided meals, another one-night assignment and the search for an apartment they can afford.

"It's the reverse of the American dream," said Tom Nees, director of the Community of Hope, which operates an 11-apartment shelter on Belmont Street. "They're trying to find less-desirable apartments so their public assistance check covers it and they don't get evicted. They're trying to climb down the ladder."

Corinthia Thompson, 22, a veteran of the open market, is temporarily living with two girlfriends and her 3-year-old son in a one-bedroom apartment in Southeast Washington. She uses bus tokens supplied by the city to hunt for an apartment that she can afford on her $257-a-month public assistance check.

"Christmas is not even on my mind this year," said Thompson, who said she has not been settled since August when her mother-in-law moved without notice from the Oxon Hill apartment they had been sharing. "I just need my own place. Then I can get stabilized. But my boy sees all the toys on TV, "Sesame Street" things, and wants them all. I wish he were younger and didn't know."

The open-market arrangement is supposed to be temporary and the inconvenience an incentive for the families to find housing on their own.

But at least two families among the 61 people on the open market have been there for two months, according to Sandra Mason, a city social worker who has been providing emergency housing since 1978. "If I had one Christmas wish, it would be for more low-income housing to be built by the city and federal government," she said. "I call the city public housing and they're working on applicants from 1976. I call Section 8 housing program and their federal money never comes in."

Last year, the city allowed families to stay indefinitely at the Capital City motel, but it found that the residents abused hotel property and rarely searched for housing. "It was too expensive," said Dennis Bethea, the newly appointed chief of the Office of Emergency Shelter and Support Services. "Frankly, we learned from that experience."

Although motel space has always been used in a crunch, the Department of Human Resources began the open-market program in earnest this summer because of increased demand, Bethea said. "Open market just reflects our inability to contract with enough facilities. Our goal is to have all families out of that type of arrangement by March."

The city tries to help establish families outside of shelters, often sending families to apartments that are smaller than they can legally occupy just to move them from the life of shelters.

"Families beg to allow them to get into one room so they can afford it," said Rita Branham, housing director for Community of Hope. Women come to her confused, she said. "They're living in cars or in a rooming house with an abusive man," she said. "We had a woman living with four children in a furnace room."

Branham's task is to find $250-a-month housing in a city where it rarely exists. Only seven city realty companies accept tenants on public assistance, she said.

As a result, each day Branham obtains the list of planned evictions in the city, for two reasons. First, she checks to see if it includes someone in the shelter's neighborhood who will need help. Then she checks it to see if one of the soon-to-be vacant apartments could be rented by one of the families in her shelter. "It's a revolving door," she said. "The only hope some people have is that their number is going to come up one day on the public housing list."

Sometimes the only thing Branham can offer the women is a private place to cry. "The women are so intimidated," she said. "They don't know how to go out there and find those apartments, even fill out the application."

Nees is particularly worried about the health of the children living in shelters and motels. Two children who were living at the Pitts Hotel have died in the last three months, one of meningitis, the other of pneumonia. "No one can say whether these children would have died if they hadn't been in temporary housing," said Nees.

Although many of those living in shelters are covered by Medicaid, the government's medical program for the poor, transportation to clinics and doctors remains a problem, he said. In recent weeks, Nees says, he has received assurances from several pediatricians in Southeast Washington that they will begin to evaluate the children living in the Community of Hope shelter.

"That's one bright spot on the horizon," he said. "But mostly, this Christmas season is just a time of stress here. There's a despair of trying to make something out of nothing."