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Cole Leinbach, a librarian in Tieton, Wash., population 1,610, watched intently as a 7-year-old girl hunkered down with a book in a corner of the town’s one-room library. Her brother, 4, had opened a board game searching for potential toys. Their mother talked quietly on her phone in Spanish. “This is what libraries are supposed to be,” he said, “just a place a mom can go with her kids for an hour to hang out and get some kind of enriching entertainment.” But the Tieton library, which occupies a few hundred square feet in a side room at the city hall, is closing next month, a casualty of rising costs in Yakima County, Wash., shrinking help from Washington, D.C., financing decisions made decades ago and significant demographic change. “I’ve had people come express dismay,” said Mr. Leinbach, who at 26 has been a librarian for about a year and a half. “A library is in a lot of ways a kind of civic symbol, a demonstration of a community’s commitment to itself. So what does it mean if that goes away?” That is a question a growing number of communities, many of them in Republican states, will be facing soon. In March, President Trump issued an executive order dismantling the Institute of Museum and Library Services, which has provided around $270 million a year to public and academic libraries to help pay for services such as summer reading programs, broadband internet access, lending between libraries, staff training and access to national databases. A court fight over that order continues, with a temporary injunction keeping the agency operating. But state and local governments, already concerned over the uncertainty of the looming library cuts, are also confronting rising personnel costs, a slowing economy and a new law that would pare back Medicaid and food assistance. Because of all that, those governments may find themselves unable to pick up the slack if federal money shrinks or goes away. Thank you for your patience while we verify access. If you are in Reader mode please exit and log into your Times account, or subscribe for all of The Times. Thank you for your patience while we verify access. Already a subscriber? Log in. Want all of The Times? Subscribe.