The weak tightness wt(Ⅹ), introduced in 6, has the property wt(Ⅹ) ≤ t(Ⅹ). It was shown in 4 that if X is a homogeneous compactum then X ≤ 2~(wt(Ⅹ)πx(Ⅹ)). We introduce the almost tightness at(Ⅹ) with the property wt(Ⅹ) ≤ at(Ⅹ) ≤ t(Ⅹ) and show that if X is a power homogeneous compactum then X ≤ 2~(at(Ⅹ)πx(Ⅹ)). This improves the result of Arhangel'skii, van Mill, and Ridderbos in 2 that X ≤ 2~(t(Ⅹ)) for a power homogeneous compactum X and gives a partial answer to a question in 4, In addition, if X is a homogeneous Hausdorff space we show that X ≤ 2~(pw_cL(Ⅹ)wt(Ⅹ)πx(Ⅹ)pct(Ⅹ)), improving a result in 3. It also extends the result in 4 into the Hausdorff setting. The cardinal invariant pwL_c(Ⅹ), introduced in 5 by Bella and Spadaro, satisfies pwL_c(Ⅹ) ≤ L(Ⅹ) and pwL_c(Ⅹ) ≤ c(Ⅹ). We also show the weight w(Ⅹ) of a homogeneous space X is bounded in various contexts using wt(Ⅹ). One such result is that if X is homogeneous and regular then w(Ⅹ) ≤ 2~(L(Ⅹ)wt(Ⅹ)pct(Ⅹ)). This generalizes a result in 4 that if X is a homogeneous compactum then w(Ⅹ) ≤ 2~(wt(Ⅹ)).
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