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Trees and Ordinal Indices in C(K) Spaces for K Countable Compact

Description: In the dissertation we study the C(K) spaces focusing on the case when K is countable compact and more specifically, the structure of C() spaces for < ω1 via special type of trees that they contain. The dissertation is composed of three major sections. In the first section we give a detailed proof of the theorem of Bessaga and Pelczynski on the isomorphic classification of C() spaces. In due time, we describe the standard bases for C(ω) and prove that the bases are monotone. In the second section we consider the lattice-trees introduced by Bourgain, Rosenthal and Schechtman in C() spaces, and define rerooting and restriction of trees. The last section is devoted to the main results. We give some lower estimates of the ordinal-indices in C(ω). We prove that if the tree in C(ω) has large order with small constant then each function in the root must have infinitely many big coordinates. Along the way we deduce some upper estimates for c0 and C(ω), and give a simple proof of Cambern's result that the Banach-Mazur distance between c0 and c = C(ω) is equal to 3.
Date: August 2015
Creator: Dahal, Koshal Raj
Partner: UNT Libraries

[Curvy Trees]

Description: Photograph of three curvy trees in the botanical gardens of Hangzhou, China. The trees are evenly spaced in the foreground. A white wall is visible behind the trees.
Date: unknown
Creator: Gough, Ray
Partner: UNT College of Visual Arts + Design

[Trees in the Snow]

Description: Photograph of a group of trees in the snow. The dark trunks stand in sharp contrast to the white snow and gray sky, and more trees can be seen in the distant background.
Date: 19uu
Creator: Clark, Joe
Partner: UNT Libraries Special Collections

[The Drive-In Outhouse]

Description: Photograph of a couple of cars and miscellaneous parts left under the shade of two large trees. Branches of one tree appear to have fallen against the fence that runs behind them. Narrative by Junebug Clark: Also in the center of the photo and in the shade of the trees is an outhouse. One of many Joe photographed for his book, "Early American Architecture."
Date: 19uu
Creator: Clark, Joe
Partner: UNT Libraries Special Collections

[Winter Scene with Bulldozer and Attached Poem]

Description: Photograph of a Bulldozer, a large tree, and one small bird in the snow with a poem talking in direct reference to them. In the image, the bulldozer sits next to a lone tree with a small bird sitting on one of the branches. The small poem is glued on top of the image on the far left,
Date: unknown
Creator: Clark, Joe
Partner: UNT Libraries Special Collections

[The Old Apply Tree]

Description: Photograph of an apple tree in the snow. On the back of the photograph the tree has been called "the old apple tree." In the high contrast image, the tree, completely black, fills the frame and is covered in snow.
Date: January 30, 1957
Creator: Clark, Joe
Partner: UNT Libraries Special Collections

[Icicles]

Description: Photograph of icicles on the edge of tree branches. In the image, the background is blown-out with the icicles almost disappearing into it, with the ice covered branches completely black. Narrative by Junebug Clark: Print was made from a negative process that Joe called "Impro" It was a method of dropping out midtones in a photographic image. Photo by: Joe Clark, HBSS.
Date: January 30, 1957
Creator: Clark, Joe
Partner: UNT Libraries Special Collections

Arbor day, its purpose and observance.

Description: Discusses Arbor Day as a holiday that recognizes the importance of United States forests and discusses a citizen's duty to preserve them and foster new growth of trees.
Date: November 1936
Creator: United States. Forest Service.
Partner: UNT Libraries Government Documents Department