Consistency results in the theory of continuous functions and selective separability

We study of the notion of selective separability (SS), which was introduced by Marion Scheepers, and its connection with the game-theoretic strengthening, strategically selective separable spaces (SS+). It is known that every set of countable pi-weight is selectively separable, and if X is selectively separable, then all dense subsets of X are selectively separable. We know that some dense countable subsets of 2c are selectively separable and some are not. It is also known that Cp(X) is selectively separable if and only if it is separable and has countable fan tightness. Here we prove that separable Frechet spaces are selectively separable. It is also shown that consistently the product of two separable Frechet spaces might not be selectively separable. Also we show that adding a Sacks real can destroy the property of being selectively separable. We introduce a notion stronger than selective separability and called it strategically selective separable or SS+ and consider its properties in countable dense subsets of uncountable powers. We show that there is an SS space which fails to be SS+. The motivation for studying SS+ is that it is a property possessed by all separable subsets of Cp(X) for each sigma-compact space X. We prove that the winning strategy for countable SS+ spaces can be chosen to be Markov. We introduce the notion of being compactlike of a collection of open sets in a topological space and with the help of this notion we prove that there are two countable SS+ spaces such that the union fails to be SS+, which contrasts the known result about the union of SS spaces. We also prove that the product of two countable SS+ spaces is again countable SS+. We prove a very interesting result which consistently contrasts our previous result, that the proper forcing axiom, PFA, implies that the product of two countable Frechet spaces is SS. Also we show that consistently with the negation of CH that all separable Frechet spaces have pi-weight at most o1. We also worked on an open question posed by Ohta and Yamasaki in the book "Open Problems in Topology" which is, whether every C*-embedded subset of a first countable space is C-embedded. It is known that a counterexample can be derived from the assumption b=s=c and that if the Product Measure Extension Theorem (PMEA) holds then the answer is affirmative in some cases. We show that in the model obtained by adding kappa many random reals, where kappa is a supercompact cardinal, every C*-embedded subset of a first countable space (even with character smaller than kappa) is C-embedded. The result was derived from the interesting fact that, if two ground model sets are completely separated after adding a random real, then they were completely separated originally. The dissertation is divided as follows. The first chapter contains the topological properties of selectively separable spaces. The second chapter contains all the results we obtained about SS+ spaces. The third chapter is devoted to the theorems involving CH and forcing extensions. The final chapter contains the results we obtained in the random real model about the C-embedding and C*-embedding properties. Any topological term not defined explicitly should be understood as in [1]. The corresponding remark applies to set theoretic notions and [2]

Matrix Rigidity Induces Osteolytic Gene Expression of Metastatic Breast Cancer Cells

Nearly 70% of breast cancer patients with advanced disease will develop bone metastases. Once established in bone, tumor cells produce factors that cause changes in normal bone remodeling, such as parathyroid hormone-related protein (PTHrP). While enhanced expression of PTHrP is known to stimulate osteoclasts to resorb bone, the environmental factors driving tumor cells to express PTHrP in the early stages of development of metastatic bone disease are unknown. In this study, we have shown that tumor cells known to metastasize to bone respond to 2D substrates with rigidities comparable to that of the bone microenvironment by increasing expression and production of PTHrP. The cellular response is regulated by Rho-dependent actomyosin contractility mediated by TGF-ß signaling. Inhibition of Rho-associated kinase (ROCK) using both pharmacological and genetic approaches decreased PTHrP expression. Furthermore, cells expressing a dominant negative form of the TGF-ß receptor did not respond to substrate rigidity, and inhibition of ROCK decreased PTHrP expression induced by exogenous TGF-ß. These observations suggest a role for the differential rigidity of the mineralized bone microenvironment in early stages of tumor-induced osteolysis, which is especially important in metastatic cancer since many cancers (such as those of the breast and lung) preferentially metastasize to bone

Six-Month Results of Treatment-Blinded Medication Titration for Hypertension Control Following Randomization to Endovascular Ultrasound Renal Denervation or a Sham Procedure in the RADIANCE-HTN SOLO Trial

The multicenter, international, randomized, blinded, sham-controlled RADIANCE-HTN SOLO trial demonstrated a 6.3 mmHg greater reduction in daytime ambulatory systolic blood pressure (BP) at 2 months by endovascular ultrasound renal denervation (RDN) compared with a sham procedure among patients not treated with antihypertensive medications. We report 6-month results following the addition of a recommended standardized stepped-care antihypertensive treatment (SSAHT) to the randomized endovascular procedure under continued blinding to initial treatment