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For metrizable continua, there exists the well-known notion of a Whitney map. Garcia-Velazquez extends the definition of Whitney map for C(X), where X is an arbitrary continuum (not necessarily metrizable). He also shows that the examples he considers do not admit such generalized Whitney maps. In this paper we shall investigate the properties of continua which admit such generalized Whitney maps.

The main purpose of this paper is to prove that some hyperspaces of a non-metric chainable continuum have the fixed point property.

The purpose of this paper is to prove that a C-continuum X is metrizable if and only if it admits a Whitney map for C(X) (Theorem 2.5).

The main purpose of this paper is to study the weight under hereditarily irreducible mappings between continua. The main result states that if f : X → Y is an hereditarily irreducible and surjective mapping of a D-continuum X, then w(X) = w(Y ).

It is known that the limit of an inverse system of totally regular continua is a totaly regular continuum. In this paper we prove that this is true for approximate limit of an approximate inverse system in the sense of S. Mardešić.

Let X be a non-metric continuum, and C(X) be the hyperspace of subcontinua of X. It is known that there is no Whitney map on the hyperspace 2X for non-metrizable Hausdorff compact spaces X. On the other hand, there exist non-metrizable continua which admit and ones which do not admit a Whitney map for C(X). In this paper we will study the propertis of non-metric spaces X which admit a Whitney map for C(X).

Let X be a non-metric continuum, and C(X) be the hyperspace of subcontinua of X. It is known that there is no Whitney map on the hyperspace 2X for non-metrizable Hausdorff compact spaces X. On the other hand, there exist non-metrizable continua which admit and ones which do not admit a Whitney map for C(X). In particular, locally connected or rim-metrizable continuum admits a Whitney map if and only if it is metrizable. In this paper we will show that an arc-smooth continuum X admits a Whitney map for C(X) if and only if it is metrizable.

The main purpose of this paper is to study the properties of continua with the property of Kelley which admit a Whitney map either for hyperspace C(X). In particular, it is proved that an arcwise connected continuum X with the property of Kelley admits a Whitney map for C(X) if and only if it is metrizable.

Let C(X) be the hyperspace of all subcontinua of a metric continuum X. Alejandro Illanes has proved that C(X) is a finite dimensional Cartesian product if and only if X is an arc or a circle. In this paper we shall prove, using the inverse systems and limits, that if X is a non-metric rim-metrizable continuum and C(X) is a finite dimensional Cartesian product, then X is a generalized arc or a generalized circle. It is also proved that if X is a non-metric continuum such that dim(X) < ∞ and such that X has the cone = hyperspace property, then X is a generalized arc, a generalized circle, or an indecomposable continuum such that each nondegenerate proper subcontinuum of X is a generalized arc.

The main purpose of this paper is to prove same theorems concerning nonmetric hereditarily locally connected continua using inverse systems.

The main purpose of this paper is to study the inverse limits of continuous image of arcs. We shall prove: a) If X = (Xa, pab, A) is a monotone well – order inverse system of contonuous image of arcs such that cf(A) not equal omega 1, then X = lim X is the continuous imege of an erc (Theorem 2.17). b) b) Let X = (Xa, pab, ( A, less or equal)) be an inverse system of continuous image of arcs with monotone surjective bonding mappings. Then X = limX is the continuous image of an arc if and only if for each cyclic element Z of X and the points x, y, z elements of Z there exists a countable direct subset (B, less or equal) of (A, less or equal) such that for each countable direct subset (C, less or equal) of (A, less or equal) with B subset of C the restriction hBC = pBC|lim(Wd(x,y,z),pdd1,D) of the canonical projection pBC is a homeomorphism (Theorem 2.22).

In this paper we define the Freudenthal space for approximate systems of compact spaces. The original construction due to Freudenthal was done for inverse sequences. We prove that our construction gives a paracompact space, which, in the case of an approximate sequence, is compact metric. The result is applied to approximate systems of continua and of hereditarily unicoherent continua, giving that the limit space is a continuum or a hereditarily unicoherent continuum, respectively.

The main purpose of this paper is the investigation of the continuity of the near-compactification. Section One contains the definition and basic properties of the near-compactification hX of a space X. Section Two is the main section. We say that the near-compactification is X-continuous if h(limX) = lim hX for some inverse system X. The necessairy and sufficient conditions for X-continuity of the near-commpactification are given.

In the present paper the following generalization of the well-known Sirota's theorem is proved. Let X = {; ; X_a, e_a, p_(ab), A}; ; be an approximate inverse system of metric compacta X_a, a € A. The spaces Exp(limX) and limExp(X) are homeomorphic.

Klasu wc-preslikavanja uveo je Harris u radu [4] i postavio pitanje interne defi­nicije tih preslikavanja tj. definicije u terminima prostora X i Y. To je interesantno zato što su wc-preslikavanja definirana tako da wf : wX—»wY bude zatvoreno. U radu su dane neke externe i interne karakterizacije wc-preslikavanja. Nužni i dovoljni uvjeti da bi f : X^ Y bilo wc-preslikavanje dani su za T2 Frechetov prostor Y (Teorem 1.13.) i E| prebrojivo kompaktan prostor (Teorem 1.15.)., Let X be a topological T| space. By wX we denote the Wallman compactifi­ cation of X. A mapping f : X—•¥ is a wc-mapping if f has an unique closed ex­tension wf ; wX—»-wY. The main purpose of this paper is to give some external and internal charac­terizations of the wc-mappings. The main theorem establishes that if Y is a Hausdorff Frechet space then f is a wc-mapping iff f is an uw-mapping (Theorem 1T3.)

Fop every Hausdorff space X the spaces XΘ and Xu are introduced. If X is H-closed (Urysohn-closed) then XΘ (Xu) is compact T1-space. If f: X->Y is a mapping, then there exist the mappings fΘ:XΘ->YΘ and fu: Xu->Yu. We say that f : X->Y is a Θ-closed (u-closed) mapping if fΘ (fu) is a closed mapping. If X and Y are H-closed (Urysohn-closed) and f: X->Y is the HJ-mapping, then fΘ(fu) is Θ-closed (u-closed) Let X = {Xα , fαβ ,A } be an inverse system of the H-closed (Urysohn-closed) spaces Xα and the Θ-closed (u-closed) mappings fαβ . If Xα are non-empty spaces, then X = limX ≠ Ø

In the present paper we prove the continuity (countable continuity) of the various kinds of connectedness. We generalize the classical results of Capel and Smith.

Relative continuity of the Čech-Stone functor β is proved.

Let X={;X_n, f_nm, N}; be an inverse sequence. In the paper it is proved that whenever X_n are non-empty countably compact spaces, then X = lim X is a non-empty countably compact space. Moreover the countable continuity of connectedness, normality and local connectedness is proved for various kinds of countably compact spaces.

In this paper we show various theorems which are partial answers to the question: Let X be a limit space of an inverse system where the spaces X_a, are < n-dimensional (n-dimensional, countable-dimensional, infinite-dimensional). Under what conditions is it true that X is an n-dimensional (countable-dimensional, infinite-dimensional) space? We also show some theorems on the limit space of an inverse system of Cantor-manifolds.

In this paper we shall investigate the inverse systems of the lightly compact spaces (pseudocompact or H-closed spaces) and the inverse systems of countably paracompact or the Lindelof spaces.

In this paper we investigate the countable-dimensionality of the inverse limit spaces. Section One contains theorems concerning the countable-dimensionality of a limit space of an inverse system with closed bonding mappings. In Section Two we give some theorems on countable-dimensionality of a limit of an inverse system with open mapping. Section Three is devoted to the inverse system with d-mappings or with a-reducible mappings.

The inverse sequences of ANRs are studied. It is proved the following Theorem 1. Let II be a family of ANR's. Then every Il-like continuum X is the inverse limit of an inverse sequence with maps onto and with continua belonging to II.