A continuumis said to be Suslinian if it does not contain uncountably many mutually exclusive non-degenerate subcontinua. Fitzpatrick and Lelek have shown that a metric Suslinian continuum $X$ has the property that the set of points at which $X$ is connected im kleinen is dense in $X$ . We extend their result to Hausdorff Suslinian continua and obtain a number of corollaries. In particular, we prove that a homogeneous, non-degenerate, Suslinian continuum is a simple closed curve and that each separable, non-degenerate, homogenous, Suslinian continuum is metrizable.

A continuumis said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight atmost ${{\omega }_{1}}$ and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set.

Let A be an m × n matrix of rank r and B an m × 1 matrix, both with integer entries. Let M2 be the maximum of the absolute values of the r × r minors of the augmented matrix (A | B). Suppose that the system A x = B has a non-trivial solution in non-negative integers. We prove (1) If r = n - 1 then the system A x = B has a non-negative non-trivial solution with entries bounded by M2. (2) If A has a r x n submatrix such that none of its r x r minors is 0 and x ≥ 0 is a solution of Ax=B in integers such that is minimal, then .

In (1) Armentrout raised the question “Is there a monotone decomposition of E3 into arcs?” The analogous question for E2 was answered negatively by Roberts in (8). Our aim in this paper is to give a partial answer to Armentrout's question by proving the following theorem.