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Coincidence theory for compact morphisms
- Donal O’Regan^{1}Email author
https://doi.org/10.1186/s13663-017-0613-0
© The Author(s) 2017
Received: 7 July 2017
Accepted: 6 September 2017
Published: 15 December 2017
Abstract
In this paper we present several coincidence type results for morphisms (fractions) in the sense of Gorniewicz and Granas.
Keywords
MSC
1 Introduction
Morphisms (Vietoris fractions) in the sense of Gorniewicz and Granas were introduced in 1981 and coincidence spaces were discussed. In this paper, using compact morphisms, we present a variety of coincidence (and fixed point) results on particular Hausdorff topological spaces. These spaces include \(\operatorname{ES}(\mathrm{compact})\), \(\operatorname{AES}(\mathrm{compact})\), general admissible and general dominated spaces. Our theory is motivated partly by ideas in [1–4].
Now we present some ideas needed in Section 2. Let H be the C̆ech homology functor with compact carriers and coefficients in the field of rational numbers K from the category of Hausdorff topological spaces and continuous maps to the category of graded vector spaces and linear maps of degree zero. Thus \(H(X)=\{H_{q}(X)\} \) (here X is a Hausdorff topological space) is a graded vector space, \(H_{q}(X) \) being the q-dimensional C̆ech homology group with compact carriers of X. For a continuous map \(f:X \to X\), \(H(f) \) is the induced linear map \(f_{\star}=\{ f_{\star q} \} \) where \(f_{\star q}:H_{q}(X) \to H_{q}(X)\). A space X is acyclic if X is nonempty, \(H_{q}(X)=0\) for every \(q \geq1\), and \(H_{0}(X)\approx K\).
- (i)
for each \(x\in X\), the set \(p^{-1}(x) \) is acyclic,
- (ii)
p is a perfect map, i.e., p is closed and for every \(x\in X\) the set \(p^{-1}(x) \) is nonempty and compact.
For a subset K of a topological space X, we denote by \(\operatorname{Cov}_{X} (K) \) the set of all coverings of K by open sets of X (usually we write \(\operatorname{Cov} (K)=\operatorname{Cov}_{X} (K)\)). Given a morphism \(\phi\in M(X,X)\) and \(\alpha\in \operatorname{Cov} (X)\), a point \(x\in X \) is said to be an α-fixed point of ϕ if there exists a member \(U\in\alpha \) such that \(x\in U \) and \(\phi(x) \cap U \neq\emptyset\). Given a morphism \(\phi\in M(X,X)\) (here \(\phi= \{ X \stackrel{p}{\Leftarrow} \Gamma \stackrel{q}{\rightarrow} X\}\)) and \(\alpha\in \operatorname{Cov} (X)\), a point \(y\in\Gamma\) is said to be an α-coincidence point for \((p,q)\) if there exists a member \(U\in\alpha \) with \(p(y) \in U\) and \(q(y) \in U\). We say ϕ has an α-coincidence if ϕ has an α-coincidence point for each representation \((p,q)\) of ϕ.
Let X and Y be topological spaces. Given two morphisms \(\phi\in M(X,Y)\) and \(\psi\in M(X,Y)\) and \(\alpha\in \operatorname{Cov} (Y)\), ϕ and ψ are said to be α-close if for any \(x\in X \), there exists \(U_{x} \in\alpha\) with \(\phi(x) \cap U_{x} \neq \emptyset\) and \(\psi(x) \cap U_{x} \neq\emptyset\). Recall that, given two single valued maps \(f, g:X \to Y \) and \(\alpha\in \operatorname{Cov} (Y)\), f and g are said to be α-close if for any \(x\in X \) there exists \(U_{x} \in\alpha\) containing both \(f(x) \) and \(g(x)\). Given a morphism \(\phi\in M(X,Y)\) and a single valued map \(g:X \to Y\) and \(\alpha\in \operatorname{Cov} (Y)\), ϕ and g are said to be strongly α-close if for any \(x\in X \) there exists \(U_{x} \in\alpha\) with \(\phi(x) \subseteq U_{x}\) and \(g(x) \in U_{x}\).
Let T be the Tychonoff cube (i.e., Cartesian product of copies of the unit interval). Finally we recall the following result from the literature; see [3] (see Theorem 7.6 and the proof of Theorem 5.5) or alternatively see [5] (see Corollary 6.5 and if we take the Hausdorff locally convex topological vector space E containing T we just need to note that T is a retract of E [6]).
Theorem 1.1
Let \(\phi\in M(T,T)\) be compact. Then ϕ has a coincidence.
2 Coincidence theory
By a space we mean a Hausdorff topological space. Let Q be a class of topological spaces. A space Y is an extension space for Q (written \(Y \in\operatorname{ES}(Q)\)) if, \(\forall X \in Q\) and \(\forall K \subseteq X \) closed in X, any continuous function \(f_{0}:K \to Y \) extends to a continuous function \(f:X \to Y\).
Theorem 2.1
Let \(X\in \operatorname{ES}(\mathrm{compact}) \) and \(\phi\in M(X,X) \) is compact. Then ϕ has a coincidence.
Proof
Let \(\phi= \{ X \stackrel{p}{\Leftarrow} \Gamma\stackrel{q}{\rightarrow} X\}: X \to X\). We know [4] that every compact space is homeomorphic to a closed subset of the Tychonoff cube T, so as a result \(K=\overline{\phi(X)}\) can be embedded as a closed subset \(K^{\star} \) of T; let \(s:K \to K^{\star} \) be a homeomorphism. Also let \(i:K \hookrightarrow X \) and \(j:K^{\star} \hookrightarrow T \) be inclusions. Now since \(X\in \operatorname{ES}(\mathrm{compact}) \) and \(i s^{-1}:K^{\star} \to X\), \(i s^{-1}\) extends to a continuous function \(h:T \to X\). Let \(\psi=j s \phi h\) and note (see [3], see (4.2)) \(\psi\in M(T,T)\) is compact. Now Theorem 1.1 guarantees that \(j s \phi h\) has a coincidence and therefore (see [3] (Lemma 6.3)) \(h j s \phi\) has a coincidence. Thus there exists a \(y\in \Gamma\) with \(h j s q(y)=p(y)\); note \(h j s \phi= \{ X \stackrel{\overline{p}}{\Leftarrow} \Gamma\boxtimes K \stackrel{\overline{q}}{\rightarrow} X\}\) where \(\Gamma\boxtimes K=\{(z_{1},z_{2}) \in\Gamma\times K: q(z_{1})=z_{2}\}\), \(\overline{p}(z_{1},z_{2})=p f_{1}(z_{1},z_{2})\), \(\overline{q}(z_{1},z_{2})=h j s f_{2}(z_{1},z_{2})\), \(f_{1}(z_{1},z_{2})=z_{1}\) and \(f_{2}(z_{1},z_{2})=z_{2}\) so \(\overline{p}(z_{1},z_{2})=p(z_{1})\) and \(\overline{q}(z_{1},z_{2})=h j s (z_{2})=h j s q(z_{1})\). Also note \(h j(z)=i s^{-1}(z)\) for \(z\in K^{\star} \) so \(h j s (w)=i(w)=w\) for \(w\in K\). Consequently \(q(y)=p(y)\), so ϕ has a coincidence point (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ, so \(C(\phi) \neq \emptyset\). □
A space Y is an approximate extension space for Q (written \(Y \in\operatorname{AES}(Q)\)) if, \(\forall \alpha\in \operatorname{Cov} (Y)\), \(\forall X \in Q\), \(\forall K \subseteq X \) closed in X, and any continuous function \(f_{0}:K \to Y\), there exists a continuous function \(f:X \to Y \) such that \(f|_{K} \) is α-close to \(f_{0}\).
Theorem 2.2
Let \(X\in \operatorname{AES}(\mathrm{compact}) \) and \(\phi\in M(X,X) \) is compact. Then for any \(\alpha\in \operatorname{Cov}_{X} (\overline{\phi(X)})\), ϕ has an α-coincidence.
Proof
Remark 2.3
One can put conditions on the space X and the morphism ϕ so that ϕ has an α-coincidence for each \(\alpha\in \operatorname{Cov}_{X} (\overline{\phi(X)})\) would guarantee that ϕ has a coincidence; for examples we refer the reader to [7] (Lemma 1.2), [3] (Lemma 6.1), [8] (Theorem 1.4 and Remark 1.6). We say \((X,\phi)\) has the α-coincidence property if ϕ, having an α-coincidence for each \(\alpha\in \operatorname{Cov}_{X} (\overline{\phi(X)})\), guarantees that ϕ has a coincidence. Thus we have: Suppose \(X\in \operatorname{AES}(\mathrm{compact})\), \(\phi\in M(X,X) \) is compact, and \((X,\phi)\) has the α-coincidence property. Then ϕ has a coincidence.
- (i)
\(g_{\alpha} \) and \(i:K \hookrightarrow W \) are α-close,
- (ii)
\(g_{\alpha}(K) \) is contained in a subset \(C_{\alpha} \subseteq W \) and \(C_{\alpha} \) has the coincidence property (i.e., any compact \(\theta\in M(C_{\alpha},C_{\alpha}) \) has a coincidence).
Theorem 2.4
- (i)
Then, for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-coincidence.
- (ii)
If \((W,\phi)\) has the α-coincidence property, then ϕ has a coincidence.
Proof
(ii). Immediate from the definition and part (i). □
- (i)
\(\psi_{\alpha} \) and \(i:K \hookrightarrow W \) are strongly α-close,
- (ii)
\(\psi_{\alpha}(K) \) is contained in a subset \(C_{\alpha} \subseteq W \) and \(C_{\alpha} \) has the coincidence property.
In our first result we will phrase it as a fixed point result but immediately after the proof we will rephrase it as a coincidence result (see Theorem 2.6).
Theorem 2.5
- (i)
Then for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-fixed point.
- (ii)
If \((W,\phi)\) has the α-fixed point property (i.e., ϕ, having an α-fixed point for each \(\alpha\in \operatorname{Cov}_{X} (\overline{\phi(X)})\), guarantees that ϕ has a fixed point), then ϕ has a coincidence.
Proof
(ii) From part (i) we know that ϕ has an α-fixed point for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\). Now since \((W,\phi)\) has the α-fixed point property, ϕ has a fixed point and, as in Section 1, ϕ has a coincidence. □
Theorem 2.6
- (i)
Then for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-coincidence.
- (ii)
If \((W,\phi)\) has the α-coincidence property, then ϕ has a coincidence.
Proof
(ii) Immediate from the definition and part (i). □
Let W be a space and C a space with the coincidence property, (i.e., any compact \(\theta\in M(C,C)\) has a coincidence). We say C dominates W if, for every compact subset K of W and every \(\alpha\in \operatorname{Cov}_{W} (K) \), there exist single valued continuous maps \(s_{\alpha}:W \to C\), \(r_{\alpha}:C \to W\) with \(r_{\alpha} s_{\alpha}: K \to W \) and \(i:K \hookrightarrow W \) α-close.
Theorem 2.7
Let W be a space and C a space with the coincidence property. Suppose \(\phi \in M(W,W) \) is compact and C dominates W. Then, for any \(\alpha \in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-coincidence.
Proof
Let W be a space and C a space with the coincidence property. We say C generally dominates W if, for every compact subset K of W and every \(\alpha\in \operatorname{Cov}_{W} (K) \), there exist \(S_{\alpha} \in M(W,C)\) and \(R_{\alpha} \in M(C,W)\) with \(R_{\alpha} S_{\alpha} \in M(K,W)\) and \(i:K \hookrightarrow W \) strongly α-close.
In our first result we will phrase it as a fixed point result but immediately after the proof we will rephrase it as a coincidence result (see Theorem 2.9).
Theorem 2.8
Let W be a space and C a space with the coincidence property. Suppose \(\phi \in M(W,W) \) is compact and C generally dominates W. Then for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-fixed point.
Proof
Theorem 2.9
Let W be a space and C a space with the coincidence property. Suppose \(\phi \in M(W,W) \) is compact and C generally dominates W. Then for any \(\alpha\in \operatorname{Cov}_{W} (\overline{\phi(W)})\), ϕ has an α-coincidence.
Proof
Follow the proof in Theorem 2.8 to obtain (2). Then there exists a \(y \in q (p^{-1}(w))\) with \(w \in q_{\alpha} (p_{\alpha}^{-1}(y))=R_{\alpha}S_{\alpha}(y)\) and, since \(R_{\alpha}S_{\alpha} \) and \(i:K \hookrightarrow W \) are strongly α-close, there exists \(U \in\alpha \) with \(R_{\alpha}S_{\alpha}(y) \in U\) and \(i(y) \in U\). Thus \(w\in U\) and \(y\in U\). Also there exists \(a\in p^{-1}(w)\) with \(y=q(a)\) and note \(p(a) =w\), i.e., \(y=q(a)\) and \(p(a)=w\). As a result \(p(a) \in U \) and \(q(a) \in U\), so ϕ has an α-coincidence (for \((p,q)\)). We can apply the above argument for any representation \((p,q)\) of ϕ. □
3 Conclusions
In this paper, using new ideas, we present a number of coincidence and α-coincidence results for compact morphisms (Vietoris fractions) defined on a variety of admissible and dominating type spaces.
Notes
Declarations
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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Authors’ Affiliations
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