>This paper deals with the relation between lattice-equivalence and some separation axioms. We are concerned with two questions: The first one is to characterize topological spaces $X$ such that $X$ and $\mathbf{F}(X)$ are lattice equivalent for some covariant functors $\mathbf{F}$ from $\mathbf{TOP}$ to itself. In the second question, it is proved that $T_{(0,2)}, T_{(S,D)}, T_{(S,1)}$ and $T_{(0,3\frac{1}{2})}$ are lattice-invariant properties but $S$, $T_{(0,1)}$, $T_{(0,S)}$, $T_{(1,2)}$, $T_{(1,S)}$, $T_{(1,3\frac{1}{2})}$, and $T_{(0,D)}$ are not.

There is a continuous bijection from $ \ell^{2}$ onto a subset of $\ell^{2}$ whose inverse is everywhere unboundedly discontinuous. If $B$ is a ball in $\ell^2$, then the continuous bijection defined on $\ell^2$ maps countably many mutually disjoint balls of $\ell^2$ into countably many mutually disjoint balls in $B$, making those images mutually disjoint.

In this paper an elementary probability question is solved and the procedure used is generalized to higher dimensions.

The generalization of designs can be considered as infinite designs. A finite set can be a part of an infinite set, a well-ordered set, a partial ordered set, or even a (discrete) topological space; in this paper designs are generalized due to this point of view. In this way for a mathematical structure ${\mathcal M}$, ${\mathcal M}$-designs are developed, especially some results in poset-designs are obtained.

In this work we show how to use the Karoubi's fundamental theorem of Hermitian K-theory [6] to prove some results in L-Theory using these same results in algebraic K-Theory.

We define and explore $\gamma$-P-regularity which is a generalization of P-regularity [16] and $\gamma$-regularity [3]. We also define and discuss strongly $\gamma$-semi-continuous functions.

We consider a one-unit system under continuous monitoring, aided by an identical spare unit and serviced by a facility that performs repair on a failed unit or preventive maintenance on a recalled unit making it as good as new. We assume instantaneous commencement of service and installation to operation. We find the distribution of the system up time and down time when life-, recall- and service-times have arbitrary probability density functions. Hence, we obtain the limiting availability of the system. Also, we compute the servicing cost per unit time to determine whether preventive maintenance is preferable over a repair only model.