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In this paper we investigate universal mappings in non-metric settings. In particular we prove that the mapping on function spaces which is induced by a monotone maping onto a locally connected non-metric continuum, is universal.

The main purpose of this paper is to study the fixed point property of non-metric tree-like continua. Using the inverse systems method, it is proved that if X is a non-metric tree-like continuum and if f : X —> X is a mapping which sends each arc component into itself, then f has the fixed point property.

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 properties of continua with the property of Kelley which admit a Whitney map either for hyperspace C(X) or for C^2(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.

Alejandro Illanes proved that if X is a metric dendrite and Y is a metric continuum such that F_2(X) is homeomorphic to F_2(Y ), then X is a dendrite. Among othe things we shall prove that this is true for a non-metric dendrite X and a non-metric locally connected continuum Y .

Let X be a topological space and 2^X the hyperspace of closed subsets of X. It is known that there is a Whitney map on the hyperspace 2^X for seperable metric spaces X. In this paper we study properties of spaces which admit a Whitney map for some subspaces of 2^X. We show that under some natural assumptions such spaces are closely related to separable metric spaces.

Ako je X = {Xg, pab, A} običan inverzni sistem T2 - prostora sa savršenim veznim preslikavanjima, tada su i projekcije p^: lim X—^X^ savršene [9, Theorem 7]. Primjerom 2.2 dokazuje se da to nije istina za aproksimativne inverzne sisteme. Glavni teorem rada (Theorem 2.1) tvrdi da su projekcije p^ savršene ako su prostori X^ lokalno kompaktni topološki kompletni (A - kompaktni) a vezna preslikavanja savršena. Spomenuti primjer 2.2 pokazuje da se lokalna kompaktnost u teoremu 2.1 ne može izostaviti. Autoru nije poznato da U se topološka kompletnost (| A| - kompaktnost) može izostaviti., The main purpose of this paper is to prove that ifK — {X^, p^h. A} is an approximate inverse system of locally compact topologically complete (^41)- compact) spaces and perfect bonding mappings, then the projections are perfect. An example which shows that the local compactness cannot he omitted is also given.

If X={X_n, p_nm, N} is a usual inverse system with confluent (monotone) bonding mappings, then the projections are confluent (monotone). This is not true for approximate inverse systems as shows Example 1.2. The main purpose of this paper is to show that the property of Kelley (smoothness) of the spaces X_n is a sufficient condition for the confluence (monotonicity) of the projections (Theorems 1.16 and 2.10).

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 characterize the continuous images of arcs by their images of the weight

The main purpose of thi paper is to prove that if $/bf X$ is a usual $/aleph_1$-directed inverse system of continuous images of arcs with monotone bonding mappings, then X = lim$/bf X$ is a continuous image of an arc.

We shall prove that if XX={;X_a, p_ab, A}; is an approximate inverse system of compact non-metric spaces with surjective bonding mappings p_ab such that each X_a is a limit of a usual t-directed inverse system XX(a)={;X_(a, g), f_(a, g)(a, e), G_a}; of metric compact spaces, then there exist: 1) a usual t-directed inverse system XX_D={;X_d, F_de, D}; whose inverse limit X_D is homeomorphic to X=lim XX ; 2) every X_dis a limit of an approximate inverse system {;X_(a, g_a), h_(a, g_a)(b, g_b), A}; of compact metric spaces X_(a, g_a) ; 3) if the mappings p_ab and f_(a, g)(a, e) are monotone, then h(a, g_a)(b, g_b) and F_de are monotone.

A characterization of continuous images of arcs by their images of weight ℵ₁is given.

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

We investigate the local connectivity of the approximate limit by means of the convergence of sets in the product. We show that the projections are monotonous, provided the spaces are normal, the product is (A)-compact and the bonding maps are monotonous. An example shows that the local connectivity of the limit of the approximate inverse system is necessary in order that the projections be monotonous whenever the bonding maps are.

The projections of an approximate system are studied.

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).

The main purpose of the present paper is to prove the following: THEOREM 3.7. Let X={Xn,Gn,Pinn,N} be an approximate inverse sequence of the dendrites.Then X=limX is hereditarily unicoherent.Moreover,if X is a Peano continuum,then X is a dendrite. THEOREM 4.4. Let X={Xn,en,pmn,N} be an inverse sequence of metric compact spaces with onto bonding mappings.If each Xn is in Class W,then /I'mX is in clas W. THEOREM 5.2. Let X={Xn,6n,PiTin,N} be an approximate inverse sequence of metric continua Xn with P(Xn) < no .Then P(X) < no, X=limX.

Sample results of this paper are the following statements: a) The inverse limit of an approximate inverse sequence of dendrites is unicoherent. Moreover, if the limit is a Peano continuum, then it is itself a dendrite. b)Consider an inverse sequence of metric compacta with onto bonding maps. If each term belongs to a class $W$ of spaces, then the limit also belongs to the class $W$.

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.

U radu je izučavana lokalna povezanost inverznog limesa inverznog sistema X= {Xa, faß, A} s potpuno zatvorenim ireducibilnim projekcijama. Prvi odjeljak sadrži glavni teorem rada, teorem L8., koji tvrdi da je limes takvog sistema s prostorima koji su lokalno povezani bez lokalnih prereznih točaka lokalno povezan. U teoremu LIO. dan je dovoljan uvjet za potpunu zatvorenost projekcija f« uz potpuno zatvorena vezna preslikavanja. U drugom odjeljku dane su primjene glavnog teorema na razne inverzne sisteme., The local connectedness of inverse limit spaces was studied in many papers ([2], [5], [7], [8], [II]). The main purpo.se of the present paper is to prove the following theorem. THEOREM 1.8 Let X={Xa, fa(i A} be an inverse system such that the projections fap are irreducible fully closed mappings. In order that limX be locally connected it is necessary that each Xa be locally connected and it is sufficient that each Xa be a locally connected space without local cut points. Key words and phrases: Locally connected spaces, closed irreducible mappings, inverse limit. Mathematch subject classification (1980): Primary 54B25; Secondary 54D05.

In this paper the non-emptines and maximal compactnes of inverse system of maximal compact spaces are proved.

Monotonicity of the projections of the approximate limit is proved.

Among other results we prove: (I) If f:X-Y is a closed surjection between normal countably compact spaces such that \Frf-l(y)\

Extensions of inverse limit space is investigated. Let P be a topological property. We say that a space Y is a P-extension of a topological space X if Y has the property P, and that there exists a subspace Z of Y such that X and Z are homeomorphic and Z is dense in Y. Let X = [X_a, f_(ab), A}; be an inverse system. In the present paper we investigate the relationship between the P-extension of the space lim X and the P-extension of the spaces X_a, aeA.

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.)

In this paper pairwise H-closed spaces and inverse bisystems are investigated. Section One contains the definition and the basic properties of PH-cIosed spaces. The various characterizations of PH-closed spaces are given.

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 it is proved that a limit of an inverse system of non-empty R-closed spaces and closed irreducible bonding mappings is a non-empty space. If X = {;X_a, f_ab, A}; is an inverse system of R-closed spaces X_a with open-closed mappings f_ab then limX is R-closed.

In the present paper we give a partial answer to the question: Let p : X -> X = {;X_a , f_ab , A}; be a resolution with dim X_a < n. Is it true that dim X < n and dam (lim X) < n. The main result of Section One is a characterization of a base of X. An important property of a resolution gives 1.8. Section Two is devoted to the mappings p_a:X-> X_a. The closedness of p_a is proved in 2.1. Section Three is the main section. The positive answer to the preceeding question is given for dim and dim_f. It is proved that a normal space X is countably compact with dim X < n iff there is a resolution p : X -> X = {; X_a, f_ab, A) such that X_a, a in A, are compact metric spaces with dim X_a < n. Section Four contains some results concerning the dimension Ind and ind.

The notion of a locally connected inverse system was introduced in the paper [4] for characterizing inverse systems of continua whose limit is locally connected. In the present paper we prove that: (1) If X is a locally connected inverse system of locally compact spaces and compact bonding mappings, then X =limX is locally connected (1.2. Theorem), (2) If X is a locally connected inverse sequence of locally countably compact spaces and pseudo-perfect bonding mappings, then X =limX is locally connected (2.1. Tbeocam) and (3) If X is a sigma-directed inverse system of regular spaces and bonding mapping with finite fibers such that X=limX is connected, then X is locally and semi-locafty connected (2.4. Theorem).

Charalambous constructed the inverse sequence of compact separable first-countable spaces X_a of the dimension Ind X_a = 1 whose inverse limit X has the dimension Ind X = 2. Among other thigs, we prove that it is impossible if the bonding mappings are dimension - raising mappings for Ind.

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.

In the present paper the invariability of the A-weak and the A-strong infinite dimensionality under a closed or open continuous mapping is investigated. The obtained results are applied to the A-weak and A-strong infinite dimensionality of the limit space of an inverse system.

Istraživanje strujnih i termičkih pojava pri prijelazu topline u vodeno-parnom traktu generatora pare

In this note it is proved among other things, that a limit of an inverse system X = (X_a, f_ab, A) is H-closed and non-empty if the spaces X_a are H-dosed and non-empty and the bonding mappings are closed irreducible surjections. Analogous theorems for an inverse system of S-closed spaces, C-com-pact spaces and NC-spaces are also proved.

U radu se izudavaju dobro uredeni inverzni sistemi X = {;X_a, f_ab, Q}; sa svojstvom hl(X_a)aleh_t+1.