Abstract
We consider the generalized ordered spaces \( X \) and \( Y \) such that the tightness \( t(X) \) coincides with the tightness \( t(Y) \) but \( T(X)=\{x\in X:t(x,X)=t(X)\} \) and \( T(Y)=\{y\in Y:t(y,Y)=t(Y)\} \) have different cardinalities. Some sufficient conditions are found under which such spaces \( X \) and \( Y \) are not \( t \)-equivalent.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1. Introduction
We consider the homeomorphism of the spaces of continuous functions on generalized ordered spaces. A space \( X \) is a generalized ordered space if \( X \) is a subspace of some linearly ordered topological space \( Y \) (see [1]). In fact, these are the linearly ordered spaces in which a neighborhood base of \( x\in X \) is given by intervals \( (y,z) \), with \( x\in(y,z) \), or by half-intervals \( [x,y) \) or \( (y,x] \) or by the singleton \( \{x\} \). Among the examples of these spaces are the Sorgenfrey line, the Michael line, the Hattori spaces, etc.
In generalized ordered spaces, the tightness \( t(X) \) of a space \( X \) coincides with the functional tightness of \( (X) \). Therefore, if \( t(X)\neq t(Y) \) for generalized ordered spaces \( X \) and \( Y \) then the spaces \( C_{p}(X) \) and \( C_{p}(Y) \) are nonhomeomorphic [2]. In the article, considering generalized ordered spaces \( X \) and \( Y \) of the same tightness but having the nonequipollent sets of the points of maximum tightness, we find the sufficient conditions for \( X \) and \( Y \) to be not \( t \)-equivalent, which means that \( C_{p}(Y) \) and \( C_{p}(X) \) are nonhomeomorphic.
As a consequence, using the homeomorphism theorems for \( C_{p}[0,\alpha] \), we obtain the topological classification of \( C_{p}(S_{\alpha}) \) on the “long Sorgenfrey lines \( S_{\alpha} \).” Note that the topological and linear classifications of \( C_{p}[0,\alpha] \) depend on the \( t \)-equivalence of the ordinal segments \( [0,\alpha] \). The theorems in this article imply that the topological classification of the spaces \( C_{p}(S_{\alpha}) \) coincides with the linear homeomorphic classification of \( C_{p}(S_{\alpha}) \) which was obtained in [3].
2. The Main Notations and Definitions
Throughout the sequel, \( {} \) and \( {} \) are the sets of reals and naturals respectively; \( \aleph_{0} \) is a countable cardinal, \( \aleph_{1} \) is the first uncountable cardinal; and \( |A| \) is the cardinality of a set \( A \). Given a generalized linear ordered space \( X \), as usual, we put
The spaces \( [x,\rightarrow) \), \( (\leftarrow,x] \), \( [y,x) \), and \( (x,y] \) are defined likewise. Observe that every linear ordered topological space \( X \) is normal (see [4, 1.7.5]) and so is every subspace \( Y\subset X \) (see [4, 2.7.5]). Therefore, all generalized ordered spaces are normal.
Definition 1
Let \( X \) be a generalized ordered space. If \( x \) is a limit point of \( (\leftarrow,x) \) then the cofinality of \( x \) is the cardinal
If \( x \) is an isolated point in \( (\leftarrow,x]\subset X \) then we put \( \operatorname{cf}x=0 \). The coinitiality of \( x\in X \) is defined similarly: \( \operatorname{cn}x=0 \) if \( x \) is an isolated point in \( [x,\rightarrow) \) and
if \( x \) is a limit point of \( (x,\rightarrow) \).
Denote by \( t(X) \) the tightness of a space \( X \) and designate as \( t(x,X) \) the tightness of \( X \) at a point \( x \). If \( X \) is a generalized ordered space, then clearly \( t(x,X)=\max\{\operatorname{cf}x,\operatorname{cn}x\} \). Let \( t(X)=\tau \). Consider the set
and its subsets
Clearly,
Definition 2
Let \( Z\subset X \) and let \( \lambda \) be a cardinal. A point \( x\in Z \) is \( \lambda \)-inaccessible in \( Z \) if from \( x\notin A\subset Z \) and \( |A|\leq\lambda \) it follows that \( x\notin\overline{A} \).
Definition 3
A function \( f:X\rightarrow{} \) is \( \lambda \)-continuous if, for every \( A\subset X \), with \( |A|\leq\lambda \), the function \( f|_{A} \) is continuous on \( A \). If for every set \( A\subset X \) such that \( |A|\leq\lambda \), there exists a continuous function \( h:X\rightarrow{} \) such that \( f|_{A}=h|_{A} \) then \( f \) is strictly \( \lambda \)-continuous.
Note that for a normal space \( X \), every \( \lambda \)-continuous function is strictly \( \lambda \)-continuous (see [5]).
It is not hard to see that the characteristic function \( \chi_{\{x\}}:X\rightarrow{} \) is \( \lambda \)-continuous if and only if \( x \) is \( \lambda \)-accessible in \( X \). As usual, \( C_{p}(X) \) means the space of all real-valued continuous functions on \( X \) which is endowed with the topology of pointwise convergence, while \( C_{\omega}(X) \) is the space of all \( \omega \)-continuous functions on \( X \) which is also endowed with the topology of pointwise convergence. The notation \( C_{p}(X)\sim C_{p}(Y) \) means that \( C_{p}(X) \) and \( C_{p}(Y) \) are homeomorphic.
Given a Tychonoff space \( X \), denote the Hewitt completion of \( X \) by \( \nu X \). The following are well known:
Theorem 1 [4]
If \( X \) and \( Y \) are Tychonoff spaces and \( \varphi:X\rightarrow Y \) is a homeomorphism then there exists a homeomorphism \( \tilde{\varphi}:\nu X\rightarrow\nu Y \) such that \( \tilde{\varphi}|_{X}=\varphi \). ☐
Theorem 2 [5]
If \( X \) is a normal space then
3. The Main Results
Lemma 1
Let \( X \) be a generalized ordered space, \( t(X)>\aleph_{0} \), and let \( P_{l}\subset T_{l}(X) \) be a right discrete space in \( T_{l}(X) \); i.e., every \( x\in P_{l} \) is isolated in \( T_{l}(X)\cap[x,\rightarrow) \). Then there exists \( A(P_{l})\subset C_{\omega}(X)\setminus C_{p}(X) \), with \( A(P_{l})=\{f_{x}:x\in P_{l}\} \), such that for every \( x\in P_{l} \) the function \( f_{x} \) is continuous on \( X\setminus\{x\} \), right continuous at \( x\in X \), and
for all \( x^{\prime},x^{\prime\prime}\in P_{l} \), \( x^{\prime}\neq x^{\prime\prime} \).
Proof
Since \( P_{l} \) is a right discrete set in \( T_{l}(X) \), for every \( x\in P_{l} \) there exists a neighborhood \( U(x) \) such that \( U(x)\cap[x,\rightarrow)\cap T_{l}(X)=\{x\} \). Putting \( f_{x}(x)=1 \) and \( f_{x}(t)=0 \) if \( t<x \) or \( t\in[x,\rightarrow)\setminus U(x) \), we extend \( f_{x} \) by continuity to \( [x,\rightarrow) \). It is not hard to see that \( f_{x} \) is left discontinuous at \( x \), continuous on \( [x,\rightarrow) \), and \( \omega \)-continuous since \( \operatorname{cf}x=\tau>\aleph_{0} \). The set \( A(P_{l})=\{f_{x}:x\in P_{l}\} \) satisfies the hypotheses of the lemma. ☐
Obviously, if \( T_{r}(X) \) contains a left discrete set \( P_{r} \); then, as in Lemma 1, we can construct \( A(P_{r})\subset C_{\omega}(X)\setminus C(X) \).
Lemma 2
Suppose that \( Y \) is a generalized ordered space, \( y_{0}\in Y \), \( \operatorname{cf}y_{0}>\aleph_{0} \), and there exists a cofinal countably compact set \( A\subset(\leftarrow,y_{0}) \). Then for every function \( f\in C_{\omega}(Y) \) there exists \( y_{f}\in Y \) such that \( y_{f}<y_{0} \) and \( f(y)=f(y_{f}) \) for every \( y\in[y_{f},y_{0}) \). The same holds for \( y_{0}\in Y \) such that \( \operatorname{cn}y_{0}>\aleph_{0} \).
Proof
Show that
(\( \ast \)) for every \( \varepsilon>0 \) there is \( y_{\varepsilon}<y_{0} \) such that \( |f(y^{\prime\prime})-f(y^{\prime})|<\varepsilon \) if \( y^{\prime},y^{\prime\prime}\in(y_{\varepsilon},y_{0}) \).
Indeed, if (\( \ast \)) fails then there is \( \varepsilon_{0}>0 \) such that for every \( y<y_{0} \) we have points \( y^{\prime},y^{\prime\prime}>y \) for which \( |f(y^{\prime})-f(y^{\prime\prime})|\geq\varepsilon_{0} \). By induction, we can choose some increasing sequence of points
such that \( \{y_{n}\}^{\infty}_{n=1}\subset A \) and \( |f(y^{\prime\prime}_{n})-f(y^{\prime}_{n})|\geq\varepsilon_{0} \) for each \( n\in{} \). Since \( A \) is countably compact, there exists \( a=\sup\{y_{n}:n\in{}\}\in A \). Clearly, \( a \) is also a limit point of \( \{y^{\prime}_{n}:n\in{}\} \) and \( \{y^{\prime\prime}_{n}:n\in{}\} \). We come to a contradiction to the continuity of \( f \) on the countable set \( \{y^{\prime}_{n}:n\in{}\}\cup\{y^{\prime\prime}_{n}:n\in{}\}\cup\{a\} \). Consequently, (\( \ast \)) holds.
Applying (\( \ast \)) to \( \varepsilon=\frac{1}{n} \), we obtain some sequence \( y_{1}<y_{2}<\cdots \) such that \( y_{n}\in A \) and \( |f(y^{\prime\prime})-f(y^{\prime})|<\frac{1}{n} \) for \( y^{\prime},y^{\prime\prime}\in(y_{n},y_{0}) \). Then for \( y_{f}=\sup\{y_{n}:n\in{}\} \) we have \( f(y)=f(y_{f}) \) for all \( y\in[y_{f},y_{0}) \). ☐
Lemma 3
Suppose that \( X \) and \( Y \) be generalized ordered sets, \( t(X)=t(Y)=\tau>\aleph_{0} \), and \( \Phi:C_{\omega}(X)\rightarrow C_{\omega}(Y) \) is a homeomorphism such that \( \Phi(C_{p}(X))=C_{p}(Y) \). If \( f_{x}\in A(P_{l}) \) (or \( f_{x}\in A(P_{r})) \) then \( \Phi(f_{x}) \) is \( \lambda \)-continuous for all \( \lambda<\tau \).
Proof
Suppose that there exists a subset \( Y_{0}\subset Y \), with \( |Y_{0}|<\tau \), and \( \Phi(f_{x})|_{Y_{0}} \) is discontinuous at some point \( y_{0}\in Y_{0} \). Given \( y\in Y_{0} \) and \( n\in N \), consider the standard neighborhoods of \( \Phi(f_{x}) \); i.e.,
Then for every \( g\in\bigcap\{U(y,n):y\in Y_{0} \), \( n\in{}\} \), we have \( g|_{Y_{0}}=\Phi(f_{x})|_{Y_{0}} \). Hence,
Since \( \Phi(C_{p}(X))=C_{p}(Y) \); therefore,
On the other hand, for every neighborhood \( \Phi^{-1}U(y,n) \) of \( f_{x} \), there exists a neighborhood
where \( F(y,n)\subset X \) is a finite subset and \( k(y,n)\in \). Obviously, \( |\bigcup\{(F(y,n):y\in Y_{0} \), \( n\in \}|<\tau \) and hence \( F=\bigcup\{F(y,n):y\in Y_{0} \), \( n\in \} \) is not cofinal to \( x \). Since \( f_{x}\in A(P_{l}) \) is \( \lambda \)-continuous for all \( \lambda<\tau \) and hence strictly \( \lambda \)-continuous (see [5]), there exists a continuous function \( h\in C_{p}(X) \) such that \( f_{x}|_{F}=h|_{F} \). Clearly,
which contradicts (1). ☐
Lemma 4
Suppose that \( X \) and \( Y \) are generalized ordered spaces, \( t(X)=t(Y)=\tau>\aleph_{0} \), and \( \Phi:C_{\omega}(X)\rightarrow C_{\omega}(Y) \) is a homeomorphism such that \( \Phi(C_{p}(X))=C_{p}(Y) \). If \( x\in T_{lr}(X) \) then \( \Phi(\chi_{\{x\}}) \) is \( \lambda \)-continuous for all \( \lambda<\tau \).
Proof
Observe that the function \( \chi_{\{x\}} \) is \( \lambda \)-continuous for \( \lambda<\tau \) provided that \( x\in T_{lr}(X) \). The rest of the proof is finished as in Lemma 3. ☐
Theorem 3
Suppose that \( X \) and \( Y \) are generalized ordered spaces, \( t(X)=t(Y)=\tau>\aleph_{0} \), \( |T(X)|>|T(Y)|\geq\aleph_{0} \), and the following are fulfilled:
\( (1) \) Either \( |T_{lr}(X)|>|T(Y)| \) or there exists a right discrete subset \( P_{l}\subset T_{l}(X) \) such that \( |P_{l}|>|T(Y)| \) or there exists a left discrete set \( P_{r}\subset T_{r}(X) \) such that \( |P_{r}|>|T(Y)| \).
\( (2) \) If \( y\in Y \) and \( \operatorname{cf}y=\tau \) then there exists a cofinal countably compact subset in \( (\leftarrow,y) \). If \( \operatorname{cn}y=\tau \) then there exists a coinitial countably compact subset in \( (y,\rightarrow) \).
Then \( C_{p}(Y) \) and \( C_{p}(X) \) are not homeomorphic.
Proof
Suppose that \( P_{l}\subset T_{l}(X) \) is right discrete in \( T_{l}(X) \) and \( |P_{l}|>|T(Y)| \). Consider
and suppose that there exists a homeomorphism \( \Phi:C_{p}(X)\rightarrow C_{p}(Y) \). Without loss of generality, we may assume that \( \Phi(0)=0 \). Using Theorems 1 and 2, we can extend \( \Phi \) to a homeomorphism \( \Phi:C_{\omega}(X)\rightarrow C_{\omega}(Y) \). Since the function \( f\equiv 0 \) is a limit point of the nonstationary sequence \( \{f_{x_{n}}\}_{n=1}^{\infty}\subset A(P_{l}) \), we see that
Since \( |T(Y)|<|P_{l}| \), there exists \( P_{l1}\subset P_{l} \) such that \( |P_{l1}|=|P_{l}| \) and \( \Phi(f_{x})(y)=0 \) for all \( x\in P_{l} \) and \( y\in T(Y) \).
Since all \( \Phi(f_{x}) \) do not belong to \( C_{p}(Y) \) and are \( \lambda \)-continuous for \( \lambda<\tau \) by Lemma 3, for every \( x\in P_{l1} \) there exists \( y\in T(Y) \) at which \( \Phi(f_{x}) \) is left discontinuous if \( y\in T_{l}(Y) \) and right discontinuous if \( y\in T_{r}(Y) \).
Let \( y\in T(Y) \) and
Since \( \bigcup\{P_{ly}:y\in T(Y)\}=P_{l1} \) and \( |P_{l1}|=|P_{l}|>T(Y) \), there exists \( y_{0}\in T(Y) \) such that \( |P_{ly_{0}}|>\aleph_{0} \). Without loss of generality, we may assume that all functions \( \{\Phi(f_{x}):x\in P_{ly_{0}}\} \) are left discontinuous at \( y_{0} \). By Lemma 2, for every \( \Phi(f_{x}) \) there exists a point \( y_{x}\in Y \) such that \( \Phi(f_{x})(y)=\Phi(f_{x})(y_{x}) \) for all \( y\in[y_{x},y_{0}) \); moreover, \( \Phi(f_{x})(y_{x})\neq 0 \) because \( \Phi(f_{x}) \) is discontinuous at \( y_{0} \) and \( \Phi(f_{x})(y_{0})=0 \). Since \( |P_{ly_{0}}|>\aleph_{0} \), for some \( n\in{} \) there is an uncountable set \( P_{ly_{0}n}=\bigl{\{}x\in P_{ly_{0}}:\Phi(f_{x})(y_{x})\geq\frac{1}{n}\bigr{\}} \).
Consider an arbitrary countable subset \( B\subset P_{ly_{0}n} \). Since \( \{y_{x}:x\in B\} \) is not cofinal to \( (\leftarrow,y_{0}) \), there exists a point \( y_{1} \) such that \( y_{x}<y_{1}<y_{0} \) for every \( x\in B \); i.e., the identically zero function is not limit point of \( \{\Phi(f_{x}):x\in B\} \). This contradicts the fact that the function \( f\equiv 0 \) is a limit point of \( \{f_{x}:x\in B\} \).
For the case of the existence of a left discrete subset \( P_{r} \) in \( T_{r}(X) \) such that \( |P_{r}|>|T(Y)| \), the proof is similar. If \( |T_{lr}(X)|>|T(Y)| \); then, instead of \( A(P_{l}) \), we must consider the set \( \{\chi_{\{x\}}:x\in T_{lr}(x)\} \) and use Lemma 4. ☐
4. On the \( t \)-Equivalence of the Spaces \( X_{\alpha} \)
Let \( X \) be a separable generalized ordered spaces and let \( \alpha \) be an ordinal. Endow the product \( [0,\alpha)\times X \) with the order relation \( (\gamma_{1},x_{1})\leq(\gamma_{2},x_{2}) \) if \( \gamma_{1}<\gamma_{2} \) or \( \gamma_{1}=\gamma_{2} \) and \( x_{1}\leq x_{2} \). Let \( B(x) \) be a neighborhood base of \( x\in X \) and let \( x_{0} \) be the first element in \( X \) (if existent). Define the neighborhood base of \( (\gamma,x)\in[0,\alpha)\times X \) as follows:
if \( x\neq x_{0} \)
if \( \gamma \) is a limit ordinal and
if \( \gamma \) is a nonlimit ordinal.
Denote the space \( [0,\alpha)\times X \) with the topology base \( \{B(\gamma,x):\gamma\in[0,\alpha),\ x\in X\} \) by \( X_{\alpha} \).
Considering the segment \( [0,\alpha] \), we obtain the space \( X_{\alpha+1} \). In particular, if \( X=[0,1)\subset{} \) and \( \alpha=\omega_{1} \) then we get a “long line,” and if \( S \) is a Sorgenfrey line with neighborhood base \( {\mathcal{B}}(x)=\{(a,x]:a<x\} \) and \( X=[0,1)\subset S \) then \( [0,\alpha)\times X=S_{\alpha} \) is a “long Sorgenfrey line.” As \( X \), we can take the “two arrows” or \( [0,1)\subset H(A) \), where \( H(A) \) is the Hattori space (see [6]). Note that the existence of the first element \( x_{0}\in X \) makes it possible to define some mapping \( \varphi(\gamma)=(\gamma,x_{0}) \) that is a homeomorphic embedding of the ordinal interval \( [0,\alpha) \) onto the closed subspace \( \{(\gamma,x_{0}):\gamma\in[0,\alpha)\}\subset[0,\alpha)\times X \). If \( \tau \) and \( \sigma \) are initial ordinals, \( \omega\leq\sigma\leq\tau \), and \( \tau \) is a regular ordinal; then \( t(X_{\tau\sigma+1})=\tau \) and \( |T(X_{\tau\sigma+1})|=|\sigma| \). It is not hard to see that \( T(X_{\tau\sigma+1}) \) has the form
is right discrete, and \( T(X_{\tau\sigma+1})=T_{l} \). (In case \( \sigma=\tau \), we must add the point \( (\tau\cdot\tau,x_{0}) \).)
The set \( \{(\tau\cdot\gamma+\beta,x_{0}):1\leq\beta<\tau\} \) is homeomorphic to the ordinal interval \( [1,\tau) \) and, hence, it is countably compact. Moreover, this is a cofinal subset of \( (\leftarrow,(\tau(\gamma+1)),x_{0}) \). Thus, \( X_{\tau\sigma+1} \) satisfy conditions (1) and (2) of Theorem 3. Consequently, we have the following
Theorem 4
Suppose that \( X \) is a separable generalized ordered space with the first element; \( \tau \), \( \lambda \), and \( \sigma \) are initial ordinals, and \( \tau \) is a regular ordinal. If \( \omega\leq\sigma<\lambda\leq\tau \) then \( C_{p}(X_{\tau\sigma+1}) \) and \( C_{p}(X_{\tau\lambda+1}) \) are nonhomeomorphic. ☐
Theorem 5
Let \( \alpha \) and \( \beta \) be infinite ordinals and let \( X \) be a separable generalized ordered space with the first element. The space \( C_{p}(X_{\alpha+1}) \) is homeomorphic to \( C_{p}(X_{\beta+1}) \) if and only if \( C_{p}[0,\alpha] \) is homeomorphic to \( C_{p}[0,\beta] \).
Proof
Consider the closed subset \( A=[0,\alpha]\times\{x_{0}\}\subset X_{\alpha+1} \) and show that there exists a continuous linear extension operator \( \Psi:C_{p}(A)\overset{{in}}{\rightarrow}C_{p}(X_{\alpha+1}) \). On every segment \( I_{\gamma}=[(\gamma,x_{0}),(\gamma+1,x_{0})] \), \( 0\leq\gamma<\alpha \), fix \( x_{1}\in X \), with \( x_{1}>x_{0} \). By Urysohn’s Lemma, there is a linear function \( g_{0}:I_{\gamma}\rightarrow[0,1] \) such that \( g_{0}([(\gamma,x_{0}),(\gamma,x_{1})])\subset\{1\} \) and \( g_{0}(\gamma+1,x_{0})=0 \). In a similar fashion, define the function \( g_{1}:I_{\gamma}\rightarrow[0,1] \), \( g_{1}([(\gamma,x_{1}),(\gamma+1,x_{0})])\subset\{1\} \), and \( g_{1}(\gamma,x_{0})=0 \). Putting \( f_{0}=\frac{g_{0}}{g_{0}+g_{1}} \) and \( f_{1}=\frac{g_{1}}{g_{0}+g_{1}} \), we obtain the partition of unity \( \{f_{0},f_{1}\} \). Consider the operator \( \Psi:C_{p}(A)\rightarrow C_{p}(X_{\alpha+1}) \) defined by the formula
if \( 0\leq\gamma<\alpha \) and \( \Psi(f)(\alpha,x)=f(\alpha,x_{0}) \). The function \( \Psi(f)|_{A} \) is equal to \( f \), whereas \( \Psi(f) \) takes values between \( f(\gamma,x_{0}) \) and \( f(\gamma+1,x_{0}) \) on \( I_{\gamma} \) and hence \( \Psi(f) \) is continuous on \( X_{\alpha+1} \). It is easy to check that \( \Psi \) is linear and continuous. In this event (see [5, 1.5]) \( C_{p}(X_{\alpha+1}) \) is linearly homeomorphic to \( C_{p}(A)\times C_{p}^{0}(X_{\alpha+1},A) \), where \( C_{p}^{0}(X_{\alpha+1},A)=\{f\in C_{p}(X_{\alpha+1}):f(A)\subset\{0\}\} \).
By the compactness of the ordinal segment \( [0,\alpha] \), the set \( \{\gamma:\sup\nolimits_{x\in X}|f(\gamma,x)|\geq\varepsilon\} \) is finite for all \( f\in C_{p}^{0}(X_{\alpha+1},A) \) and \( \varepsilon>0 \). Therefore, \( C_{p}^{0}(X_{\alpha+1},A) \) is linearly homeomorphic to the space \( \big{(}\prod\nolimits_{0\leq\gamma\leq\alpha}C_{p}^{0}(I_{\gamma})\big{)}_{c_{0}} \) defined as follows:
where
if \( 0\leq\gamma<\alpha \) and
Since all \( I_{\gamma} \)’s, with \( 0\leq\gamma<\alpha \), are homeomorphic to \( I_{0} \) and \( I_{\alpha} \) is homeomorphic to \( X \), the space
is linearly homeomorphic to
Suppose that \( C_{p}[0,\alpha] \) is homeomorphic to \( C_{p}[0,\beta] \). Clearly, in this case \( |\alpha|=|\beta| \). Since \( A \) is homeomorphic to \( [0,\alpha] \), we obtain
If \( C_{p}[0,\alpha] \) is not homeomorphic to \( C_{p}[0,\beta] \) then this means that (see [7, 8]) either
(a) \( |\alpha|\neq|\beta| \)
or
(b) \( |\alpha|=|\beta|=|\tau| \), where \( \tau \) is an initial regular ordinal and there exist initial ordinals \( \sigma,\lambda,\sigma<\lambda\leq\tau \) such that \( \tau\sigma\leq\alpha<\tau\sigma^{+} \) and \( \tau\lambda\leq\beta<\tau\lambda^{+} \).
In case (a), granted the separability of \( X \), we obtain
and so \( C_{p}(X_{\alpha+1}) \) and \( C_{p}(X_{\beta+1}) \) are nonhomeomorphic.
In case (b), \( C_{p}[0,\alpha]\sim C_{p}[0,\tau\sigma] \) and \( C_{p}[0,\beta]\sim C_{p}[0,\tau\lambda] \); therefore, by the above,
and, respectively, \( C_{p}(X_{\beta+1})\sim C_{p}(X_{\tau\lambda+1}) \). By Theorem 4, we conclude that \( C_{p}(X_{\alpha+1}) \) and \( C_{p}(X_{\beta+1}) \) are nonhomeomorphic. ☐
Remark
If \( m,n\in{} \) and \( m\neq n \); then, essentially repeating the proof in [9], we can prove that \( C_{p}(X_{\tau n+1}) \) and \( C_{p}(X_{\tau m+1}) \) are nonhomeomorphic.
If \( \sigma=n\in{} \) and \( \omega\leq\lambda<\tau \) then \( C_{p}(X_{\tau\sigma+1}) \) is nonhomeomorphic to its square, whereas \( C_{p}(X_{\tau\lambda+1}) \) is homeomorphic to its square by Theorem 5. Therefore, \( C_{p}(X_{\tau\sigma+1}) \) and \( C_{p}(X_{\tau\lambda+1}) \) are nonhomeomorphic.
Corollary 6
Let \( \alpha \) and \( \beta \) be infinite ordinals and let \( S_{\alpha+1} \) and \( S_{\beta+1} \) be “long Sorgenfrey lines.” The spaces \( C_{p}(S_{\alpha+1}) \) and \( C_{p}(S_{\beta+1}) \) are homeomorphic if and only if so are \( C_{p}[0,\alpha] \) and \( C_{p}[0,\beta] \). ☐
References
Faber M.J., Metrizability in Generalized Ordered Spaces, Mathematical Centrum, Amsterdam (1974) (Mathematical Centre Tracts; vol. 53).
Arkhangelskii A.V., Topological Function Spaces, Moscow University, Moscow (1989) [Russian].
Trofimenko N. N. and Khmyleva T. E., “Linear homeomorphisms of spaces of continuous functions on long Sorgenfrey lines,” Sib. Math. J., vol. 57, no. 3, 558–56 (2016).
Engelking R., General Topology, Heldermann, Berlin (1989).
Tkachuk V., A Cp-Theory Problem Book, Springer, New York (2015).
Chatyrko V.A. and Hattori Y., “A poset of topologies on the set of real numbers,” Comment. Math. Univ. Carolin., vol. 54, no. 2, 189–196 (2013).
Genze L.V., Gulko S.P., and Khmyleva T.E., “Classification of spaces of continuous functions on ordinals,” Comment. Math. Univ. Carolin., vol. 59, no. 3, 365–370 (2018).
Gorak R., “Functional spaces on ordinals,” Comment. Math. Univ. Carolin., vol. 46, no. 1, 93–103 (2005).
Gulko S.P., “Spaces of continuous functions on ordinals and ultrafilters,” Math. Notes, vol. 47, no. 4, 329–334 (1990).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 2, pp. 441–448. https://doi.org/10.33048/smzh.2023.64.214
Rights and permissions
About this article
Cite this article
Trofimenko, N.N., Khmyleva, T.E. On the \( t \)-Equivalence of Generalized Ordered Sets. Sib Math J 64, 424–430 (2023). https://doi.org/10.1134/S0037446623020143
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446623020143
Keywords
- ordered topological spaces
- cofinal subsets
- regular ordinals
- tightness
- functional tightness
- Hewitt completion
- homeomorphism
- topology of pointwise convergence