Daria Michalik

# Daria Michalik: Blocking properties of the diagonal in Cartesian product

10/11/22 18:54

Tuesday, November 15, 2022 17:00

[1] J. Bobok, P. Pyrih and B. Vejnar, Non-cut, shore and non-block points in continua, Glas. Mat. Ser. III 51 (71) (2016), 237–253.

*Location:*room C11-3.11*Daria Michalik (Jan Kochanowski University in Kielce)*

Speaker:Speaker:

*Title*: Blocking properties of the diagonal in Cartesian product*Abstract*: In [1], the authors present six kinds of blocking properties for points in continua. We can consider the same properties for subcontinua. During my talk I will present some results concerning the blocking properties of the diagonal in Cartesian product. Among others, I will show a new characterisation of the interval.[1] J. Bobok, P. Pyrih and B. Vejnar, Non-cut, shore and non-block points in continua, Glas. Mat. Ser. III 51 (71) (2016), 237–253.

# Daria Michalik: Symmetric products as cones

08/01/19 21:21

Tuesday, January 8, 2019 17:15

For a continuum \(X\), let \(F_n(X)\) be the hyperspace of all nonempty subsets of \(X\)with at most \(n\)-points. The space \(F_n(X)\) is called the n'th-symmetric product.

In [1] it was proved that if \(X\)is a cone, then its hyperspace \(F_n(X)\) is also a cone.

During my talk I will discuss the converse problem. I will prove that if \(X\)is a locally connected curve, then the following conditions are equivalent:

[1] A. Illanes, V. Martinez-de-la-Vega, Symmetric products as cones, Topology Appl. 228 (2017), 36–46.

*Room:*D1-215*Daria Michalik*

Speaker:Speaker:

*Title*: Symmetric products as cones*Abstract*. (join work with Alejandro Illanes and Veronica Martinez-de-la-Vega)For a continuum \(X\), let \(F_n(X)\) be the hyperspace of all nonempty subsets of \(X\)with at most \(n\)-points. The space \(F_n(X)\) is called the n'th-symmetric product.

In [1] it was proved that if \(X\)is a cone, then its hyperspace \(F_n(X)\) is also a cone.

During my talk I will discuss the converse problem. I will prove that if \(X\)is a locally connected curve, then the following conditions are equivalent:

- \(X\)is a cone,
- \(F_n(X)\) is a cone for some \(n\ge 2\),
- \(F_n(X)\) is a cone for each \(n\ge 2\).

[1] A. Illanes, V. Martinez-de-la-Vega, Symmetric products as cones, Topology Appl. 228 (2017), 36–46.

# Daria Michalik: Degree of homogeneity of connes over locally connected curves

16/11/16 09:14

Tuesday, November 22, 2016 17:15

\(X\) is \(1/n\)-homogeneous if \(X\) has exactly \(n\) orbits. In such a case we say that the degree of homogeneity of \(X\) equals \(n\). P. Pellicer Covarrubias, A. Santiago-Santos calculated the degree of homogeneity of connes over local dendrites depending on the degree of homogeneity of their bases. We will generalize above result on connes over locally connected curves.

*Room:*D1-215*Daria Michalik*

Speaker:Speaker:

*Title*: Degree of homogeneity of connes over locally connected curves*Abstract*. \(\mathcal{H}(X)\) denotes the group of self-homeomorphisms of \(X\). An orbit of a point \(x_0\) in \(X\)is the set: \(\mathcal{O}_X(x_0) = \{h(x_0) : h\in\mathcal{H}(X)\}.\)\(X\) is \(1/n\)-homogeneous if \(X\) has exactly \(n\) orbits. In such a case we say that the degree of homogeneity of \(X\) equals \(n\). P. Pellicer Covarrubias, A. Santiago-Santos calculated the degree of homogeneity of connes over local dendrites depending on the degree of homogeneity of their bases. We will generalize above result on connes over locally connected curves.